A Math

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H1 Math

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IP Math

<p>The following Venn diagram shows a sample space \(U\) and events \(A\) and \(B\). It is known that&nbsp;\(n(U)=36\), \(n(A)=11\), \(n(B)=6\), and \(n((A\cup B)\prime)=21\).</p>
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<ol class="abc-list"><li data-list-item-id="ibsl-q101-part-1-010" data-list-item-path="ibsl-q101-part-1-010"><div class="list-item-paragraph"><p>On the diagram, shade the region \((A ∪ B)′\).</p></div></li><li data-list-item-id="ibsl-q101-part-2-011" data-list-item-path="ibsl-q101-part-2-011"><div class="list-item-paragraph">Find</div><ol class="i-list"><li data-list-item-id="ibsl-q101-part-2-1-012" data-list-item-path="ibsl-q101-part-2-011/ibsl-q101-part-2-1-012"><div class="list-item-paragraph"><p>\(n(A ∩ B)\);</p></div></li><li data-list-item-id="ibsl-q101-part-2-2-013" data-list-item-path="ibsl-q101-part-2-011/ibsl-q101-part-2-2-013"><div class="list-item-paragraph">\(P(A ∩ B)\).</div></li></ol></li><li data-list-item-id="ibsl-q101-part-3-014" data-list-item-path="ibsl-q101-part-3-014"><div class="list-item-paragraph"><p>Explain why events \(A\) and \(B\) are not mutually exclusive.</p></div></li></ol>
<p><span class="inline-marks">[4]</span></p>

The following Venn diagram shows a sample space \(U\) and events \(A\) and \(B\). It is known that \(n(U)=36\), \(n(A)=11\), \(n(B)=6\), and \(n((A\cup B)\prime)=21\).

Set Theory and Probability — Combined Problems

Mutually Exclusive Events

Shade a Venn complement, calculate an intersection and explain non-mutual-exclusivity.

Probability

<p>The Venn diagram below shows information about \(120\) students in a school. Of these, \(40\) study Chinese (C), \(35\) study Japanese (J), and \(30\) study Spanish (S). A student is chosen at random from the group. Find the probability that the student</p>
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<ol class="abc-list"><li data-list-item-id="ibsl-q050-part-1-016" data-list-item-path="ibsl-q050-part-1-016"><div class="list-item-paragraph">studies exactly two of these languages;<span class="inline-marks">[1]</span></div></li><li data-list-item-id="ibsl-q050-part-2-017" data-list-item-path="ibsl-q050-part-2-017"><div class="list-item-paragraph">studies only Japanese;<span class="inline-marks">[2]</span></div></li><li data-list-item-id="ibsl-q050-part-3-018" data-list-item-path="ibsl-q050-part-3-018"><div class="list-item-paragraph">does not study any of these languages.<span class="inline-marks">[3]</span></div></li></ol>

The Venn diagram below shows information about \(120\) students in a school. Of these, \(40\) study Chinese (C), \(35\) study Japanese (J), and \(30\) study Spanish (S). A student is chosen at random from the group. Find the probability that the student

Read a three-set Venn diagram and calculate exact-two, only-one and neither probabilities.

<p>The Venn diagram below shows events A and B where&nbsp;\(P(A) = 0.3\), \(P(A ∪ B) = 0.6\) and \(P(A ∩ B) = 0.1\). The values \(m, n, p\) and \(q\) are probabilities.</p>
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<ol class="abc-list"><li data-list-item-id="ibsl-q003-part-1-014" data-list-item-path="ibsl-q003-part-1-014"><ol class="i-list"><li data-list-item-id="ibsl-q003-part-1-1-015" data-list-item-path="ibsl-q003-part-1-014/ibsl-q003-part-1-1-015"><div class="list-item-paragraph"><p>Write down the value of \(n\).</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="ibsl-q003-part-1-2-016" data-list-item-path="ibsl-q003-part-1-014/ibsl-q003-part-1-2-016"><div class="list-item-paragraph"><p>Find the value of \(m\), of \(p\), and of \(q\).</p></div></li></ol></li><li data-list-item-id="ibsl-q003-part-2-017" data-list-item-path="ibsl-q003-part-2-017"><div class="list-item-paragraph"><p>Find \(P(B')\).</p><span class="inline-marks">[2]</span></div></li></ol>

The Venn diagram below shows events A and B where \(P(A) = 0.3\), \(P(A ∪ B) = 0.6\) and \(P(A ∩ B) = 0.1\). The values \(m, n, p\) and \(q\) are probabilities.

Complementary Events

Complete Venn-region probabilities and calculate a complement.

<p>The diagram shows a circle divided into three sectors A, B and C with central angles \(90^\circ\), \(120^\circ\) and \(150^\circ\), respectively. Sectors A and B are shaded. The arrow is spun and cannot land on a boundary. Let&nbsp;\(A\), \(B\), \(C\) and \(S\) denote landing in sector A, B, C and a shaded region, respectively.</p>
<div class="graph-block" data-graph-config="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" data-width="330" data-height="300" style="width:330px;height:300px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="abc-list"><li data-list-item-id="ibsl-q077-part-1-013" data-list-item-path="ibsl-q077-part-1-013"><div class="list-item-paragraph">\(P(B)\);</div></li><li data-list-item-id="ibsl-q077-part-2-014" data-list-item-path="ibsl-q077-part-2-014"><div class="list-item-paragraph">\(P(S)\);</div></li><li data-list-item-id="ibsl-q077-part-3-015" data-list-item-path="ibsl-q077-part-3-015"><div class="list-item-paragraph">\(P(A|S)\).</div></li></ol>
<p><span class="inline-marks">[6]</span></p>

The diagram shows a circle divided into three sectors A, B and C with central angles \(90^\circ\), \(120^\circ\) and \(150^\circ\), respectively. Sectors A and B are shaded. The arrow is spun and cannot land on a boundary. Let \(A\), \(B\), \(C\) and \(S\) denote landing in sector A, B, C and a shaded region, respectively.

Probability of Single Events — General

Calculate spinner-sector, shaded-region and conditional probabilities.

<p>Consider the events \(A\) and \(B\), where&nbsp;\(P(A)=\frac25\), \(P(B^{\prime})=\frac14\) and \(P(A\cup B)=\frac78\).</p>
<ol class="abc-list"><li data-list-item-id="ibsl-q044-part-1-003" data-list-item-path="ibsl-q044-part-1-003"><div class="list-item-paragraph">Write down \(P(B)\).</div></li><li data-list-item-id="ibsl-q044-part-2-004" data-list-item-path="ibsl-q044-part-2-004"><div class="list-item-paragraph">Find \(P(A ∩ B)\).</div></li><li data-list-item-id="ibsl-q044-part-3-005" data-list-item-path="ibsl-q044-part-3-005"><div class="list-item-paragraph">Find \(P(A | B)\).</div></li></ol>
<p><span class="inline-marks">[6]</span></p>

Consider the events \(A\) and \(B\), where \(P(A)=\frac25\), \(P(B^{\prime})=\frac14\) and \(P(A\cup B)=\frac78\).

Use complements, unions, intersections and conditional probability for two events.

<p>Kai and Lina play a sequence of rounds. Kai wins the first round with probability \(\dfrac{3}{5}\). For every round after the first, the conditional probability that the winner of the preceding round also wins the next round is \(\dfrac{2}{3}\). The match ends as soon as a player wins two consecutive rounds.</p>
<ol class="i-list"><li data-list-item-id="gp-q2-i" data-list-item-path="gp-q2-i"><div class="list-item-paragraph">The first four possible winning sequences for Kai are \(KK\), \(LKK\), \(KLKK\) and \(LKLKK\), where&nbsp;\(K\) denotes a round won by Kai and \(L\) denotes a round won by Lina. Find the probability of each sequence.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="gp-q2-ii" data-list-item-path="gp-q2-ii"><div class="list-item-paragraph">Hence, using the sum to infinity of a geometric progression, find the probability that Kai wins the match.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="gp-q2-iii" data-list-item-path="gp-q2-iii"><div class="list-item-paragraph">Given that Lina wins the first round, find the probability that Kai wins the match.<span class="inline-marks">[2]</span></div></li></ol>

Kai and Lina play a sequence of rounds. Kai wins the first round with probability \(\dfrac{3}{5}\). For every round after the first, the conditional probability that the winner of the preceding round also wins the next round is \(\dfrac{2}{3}\). The match ends as soon as a player wins two consecutive rounds.

Probability - General

Problems with Infinite Outcomes

Probability of a Sequence

Geometric Progression

Sum to infinity

APGP

<p>Ava and Ben take turns throwing a dart at a target. The probability that Ava hits the target on any throw is \(\dfrac{2}{5}\), while the probability that Ben hits it on any throw is \(\dfrac{1}{3}\). All throws are independent. A player wins as soon as that player hits the target. Ava throws first.</p>
<ol class="i-list"><li data-list-item-id="gp-q1-i" data-list-item-path="gp-q1-i"><div class="list-item-paragraph">Show that the probability that both players miss in one pair of consecutive throws, one by Ava followed by one by Ben, is \(\dfrac{2}{5}\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="gp-q1-ii" data-list-item-path="gp-q1-ii"><div class="list-item-paragraph">Find the probability that Ava wins the game.<span class="inline-marks">[3]</span></div></li></ol>
<p>In a second game, a biased coin is used to choose who throws first. Ben is twice as likely as Ava to throw first. The same hit probabilities and rules apply.</p>
<ol class="i-list" start="3"><li data-list-item-id="gp-q1-iii" data-list-item-path="gp-q1-iii"><div class="list-item-paragraph">Find the probability that Ava wins the second game.<span class="inline-marks">[3]</span></div></li></ol>

Ava and Ben take turns throwing a dart at a target. The probability that Ava hits the target on any throw is \(\dfrac{2}{5}\), while the probability that Ben hits it on any throw is \(\dfrac{1}{3}\). All throws are independent. A player wins as soon as that player hits the target. Ava throws first.

In a second game, a biased coin is used to choose who throws first. Ben is twice as likely as Ava to throw first. The same hit probabilities and rules apply.

<p>The events \(A\) and \(B\) satisfy \(P(A)=\dfrac{2}{5}\), \(P(B')=\dfrac{3}{4}\) and \(P(B\mid A)=\dfrac{1}{4}\).</p>
<ol class="i-list"><li data-list-item-id="prob-b-item-i" data-list-item-path="prob-b-item-i"><div class="list-item-paragraph">Find \(P(A'\cap B')\) and hence prove that the events \(A'\) and \(B'\) are independent.<span class="inline-marks">[4]</span></div></li></ol>
<p>A third event \(C\) satisfies \(P(C)=\dfrac{1}{2}\) and \(P(A\cap B\cap C)=\dfrac{1}{20}\). It is also known that&nbsp;\(B\) and \(C\) are independent, while \(A\) and \(C\) are not independent.</p>
<ol class="i-list" start="2"><li data-list-item-id="prob-b-item-ii" data-list-item-path="prob-b-item-ii"><div class="list-item-paragraph">Find the range of possible values of \(P(A'\cap B'\cap C')\).<span class="inline-marks">[5]</span></div></li></ol>

The events \(A\) and \(B\) satisfy \(P(A)=\dfrac{2}{5}\), \(P(B')=\dfrac{3}{4}\) and \(P(B\mid A)=\dfrac{1}{4}\).

A third event \(C\) satisfies \(P(C)=\dfrac{1}{2}\) and \(P(A\cap B\cap C)=\dfrac{1}{20}\). It is also known that \(B\) and \(C\) are independent, while \(A\) and \(C\) are not independent.

Solving problems using Formula

Problems involving Inequality

<p>For events \(A\) and \(B\), it is given that&nbsp;\(P(A)=\dfrac{1}{4}\), \(P(A\cup B)=\dfrac{5}{8}\) and \(P(A\cap B')=\dfrac{1}{8}\).</p>
<ol class="i-list"><li data-list-item-id="prob-a-item-i" data-list-item-path="prob-a-item-i"><div class="list-item-paragraph">Find \(P(B)\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="prob-a-item-ii" data-list-item-path="prob-a-item-ii"><div class="list-item-paragraph">Determine, with a reason, whether the events \(A\) and \(B\) are independent.<span class="inline-marks">[2]</span></div></li></ol>
<p>For a third event \(C\), it is given that&nbsp;\(P(C)=\dfrac{2}{5}\), \(P(A\cap B\cap C)=\dfrac{1}{20}\), and \(C\) is independent of \(A\).</p>
<ol class="i-list" start="3"><li data-list-item-id="prob-a-item-iii" data-list-item-path="prob-a-item-iii"><div class="list-item-paragraph">Find \(P(A'\cap C)\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="prob-a-item-iv" data-list-item-path="prob-a-item-iv"><div class="list-item-paragraph">Find the range of possible values of \(P(A'\cap B'\cap C')\).<span class="inline-marks">[3]</span></div></li></ol>

For events \(A\) and \(B\), it is given that \(P(A)=\dfrac{1}{4}\), \(P(A\cup B)=\dfrac{5}{8}\) and \(P(A\cap B')=\dfrac{1}{8}\).

For a third event \(C\), it is given that \(P(C)=\dfrac{2}{5}\), \(P(A\cap B\cap C)=\dfrac{1}{20}\), and \(C\) is independent of \(A\).

<p>A company buys \(p\%\) of its electronic components from supplier \(A\) and the remaining \((100 - p)\%\) from supplier \(B\). The probability that a randomly chosen component supplied by \(A\) is faulty is \(0.05\). The probability that a randomly chosen component supplied by \(B\) is faulty is \(0.03\).</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;\(p = 25\), find the probability that a randomly chosen component is faulty.</span></div></li><li><div class="list-item-paragraph">For a general value of \(p\), the probability that a randomly chosen component that is faulty was supplied by \(A\) is denoted by \(\mathrm{f}(p)\). Show that&nbsp;\(\mathrm{f}(p) = \frac{0.05p}{0.02p + 3}\). Prove by differentiation that&nbsp;\(\mathrm{f}\) is an increasing function for&nbsp;\(0 \leq p \leq 100\), and explain what this statement means in the context of the question.</div></li></ol>

A company buys \(p\%\) of its electronic components from supplier \(A\) and the remaining \((100 - p)\%\) from supplier \(B\). The probability that a randomly chosen component supplied by \(A\) is faulty is \(0.05\). The probability that a randomly chosen component supplied by \(B\) is faulty is \(0.03\).

Given that \(p = 25\), find the probability that a randomly chosen component is faulty.

N2009 P2 Q7

<p>A bag contains \(10\) orange-flavoured, \(14\) strawberry-flavoured, and \(16\) cherry-flavoured sweets which are of identical shapes and sizes. Benny selects a sweet at random from the bag. If it is not cherry-flavoured he replaces it and selects another sweet at random. He repeats the process until he obtains a cherry-flavoured sweet.</p>
<p>Calculate the probability that</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the first sweet selected is strawberry-flavoured and the fourth sweet is orange-flavoured,</span></div></li><li><div class="list-item-content"><span class="list-item-text">he selects an even number of sweets.</span></div></li></ol>

A bag contains \(10\) orange-flavoured, \(14\) strawberry-flavoured, and \(16\) cherry-flavoured sweets which are of identical shapes and sizes. Benny selects a sweet at random from the bag. If it is not cherry-flavoured he replaces it and selects another sweet at random. He repeats the process until he obtains a cherry-flavoured sweet.

the first sweet selected is strawberry-flavoured and the fourth sweet is orange-flavoured,

<p>Players \(A\) and \(B\) compete in a racquet match consisting of at most \(3\) sets. Each set is won by either Player \(A\) or \(B\), and the match is won by the first person to win two sets. Player \(A\) has a probability of \(\frac{2}{3}\) of winning each of the first two sets. If the match goes into the third set, Player \(A\) has a probability of \(p\) of winning this set.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">With the aid of a tree diagram, find in terms of \(p\), the probability that player \(A\) will win the match.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Deduce the range of values of the probability that player \(A\) wins the match.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that the value of \(p\) in order for the match to be fair is \(\frac{1}{8}\).</span></div></li></ol>
<p>Given that Player \(A\) wins the match, find, in terms of \(p\), the probability that he won the \(2^{\text{nd}}\) set.</p>

Players \(A\) and \(B\) compete in a racquet match consisting of at most \(3\) sets. Each set is won by either Player \(A\) or \(B\), and the match is won by the first person to win two sets. Player \(A\) has a probability of \(\frac{2}{3}\) of winning each of the first two sets. If the match goes into the third set, Player \(A\) has a probability of \(p\) of winning this set.

With the aid of a tree diagram, find in terms of \(p\), the probability that player \(A\) will win the match.

Deduce the range of values of the probability that player \(A\) wins the match.

Show that the value of \(p\) in order for the match to be fair is \(\frac{1}{8}\).

Given that Player \(A\) wins the match, find, in terms of \(p\), the probability that he won the \(2^{\text{nd}}\) set.

2011 NYJC Prelim

<p>Box \(A\) contains \(6\) balls numbered \(1, 2, 2, 2, 5, 7\).</p>
<p>Box \(B\) contains \(4\) balls numbered \(1, 4, 4, 7\).</p>
<p>Box \(C\) contains \(3\) balls numbered \(3, 4, 6\).</p>
<p>One ball is removed at random from each box.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that the probability that each of the three numbers obtained is greater than \(3\) is \(\frac{1}{6}\).</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that each of the three numbers obtained is greater than \(3\) or the sum of the three numbers obtained is \(13\) (or both).</span><span class="inline-marks">[4]</span></div></li></ol>

Box \(A\) contains \(6\) balls numbered \(1, 2, 2, 2, 5, 7\).

Box \(B\) contains \(4\) balls numbered \(1, 4, 4, 7\).

Box \(C\) contains \(3\) balls numbered \(3, 4, 6\).

One ball is removed at random from each box.

Show that the probability that each of the three numbers obtained is greater than \(3\) is \(\frac{1}{6}\).

Find the probability that each of the three numbers obtained is greater than \(3\) or the sum of the three numbers obtained is \(13\) (or both).

<p>Two events \(A\) and \(B\) are such that&nbsp;\(\mathrm{P}(A) = p\) and \(\mathrm{P}(B) = \frac{5}{4}p\). It is given that&nbsp;\(\mathrm{P}(A \cup B) = \mathrm{P}(A|B) = 0.6\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the value of \(p\).</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain whether the events \(A'\) and \(B\) are independent.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>The events \(C\) and \(D\) are such that&nbsp;\(\mathrm{P}(C) = 0.55\), \(\mathrm{P}(D) = 0.6\) and \(\mathrm{P}(A \cap C) = \mathrm{P}(A \cap D) = \mathrm{P}(C \cap D) = q\).</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the value of \(q\) that gives the minimum value of \(\mathrm{P}(A \cup C \cup D)\) and state the minimum value of \(\mathrm{P}(A \cup C \cup D)\).</span><span class="inline-marks">[2]</span></div></li></ol>

Two events \(A\) and \(B\) are such that \(\mathrm{P}(A) = p\) and \(\mathrm{P}(B) = \frac{5}{4}p\). It is given that \(\mathrm{P}(A \cup B) = \mathrm{P}(A|B) = 0.6\).

Explain whether the events \(A'\) and \(B\) are independent.

The events \(C\) and \(D\) are such that \(\mathrm{P}(C) = 0.55\), \(\mathrm{P}(D) = 0.6\) and \(\mathrm{P}(A \cap C) = \mathrm{P}(A \cap D) = \mathrm{P}(C \cap D) = q\).

Find the value of \(q\) that gives the minimum value of \(\mathrm{P}(A \cup C \cup D)\) and state the minimum value of \(\mathrm{P}(A \cup C \cup D)\).

2025 VJC P2 Q6

<p>For events \(A\), \(B\) and \(C\), it is given that&nbsp;\(\mathrm{P}(A) = 0.3\), \(\mathrm{P}(B) = 0.4\), \(\mathrm{P}(C) = 0.42\), \(\mathrm{P}(A' \cap B \cap C) = x\) and \(\mathrm{P}(A \cap B \cap C) = 0.02\). It is also given that events \(A\) and \(B\) are independent, and that events \(A\) and \(C\) are independent.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find \(\mathrm{P}(A \cap B)\) and \(\mathrm{P}(A \cap C)\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Draw a Venn diagram to represent this situation, showing the probability in each of the eight regions, in terms of \(x\) where necessary.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the greatest and least possible values of \(\mathrm{P}(A' \cap B' \cap C)\).</span><span class="inline-marks">[2]</span></div></li></ol>

For events \(A\), \(B\) and \(C\), it is given that \(\mathrm{P}(A) = 0.3\), \(\mathrm{P}(B) = 0.4\), \(\mathrm{P}(C) = 0.42\), \(\mathrm{P}(A' \cap B \cap C) = x\) and \(\mathrm{P}(A \cap B \cap C) = 0.02\). It is also given that events \(A\) and \(B\) are independent, and that events \(A\) and \(C\) are independent.

Find \(\mathrm{P}(A \cap B)\) and \(\mathrm{P}(A \cap C)\).

Draw a Venn diagram to represent this situation, showing the probability in each of the eight regions, in terms of \(x\) where necessary.

Find the greatest and least possible values of \(\mathrm{P}(A' \cap B' \cap C)\).

2025 TMJC P2 Q6

<p>In a simple archery game, a player shoots an arrow at a target. The probability of hitting the target depends on the player’s skill and the wind conditions for the day. The wind can either be good or bad, with the probability of a good wind day being \(0.7\).</p>
<p>According to past statistics:</p>
<ul class="bullet-list"><li><div class="list-item-content"><span class="list-item-text">Dave has a \(70\)% chance of hitting the target on a good wind day and a \(60\)% chance on a bad wind day.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Rafael has an \(80\)% chance of hitting the target on a good wind day and a \(50\)% chance on a bad wind day.</span></div></li></ul>
<p>Dave and Rafael decide to compete in a game where each player shoots \(2\) arrows at the target. The player who hits the target more times wins the game.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that Dave wins the game on a good wind day.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>It is now known that the probability Dave wins the game on a bad wind day is \(0.39\).</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Given that Dave wins the game, find the probability that the day was a bad wind day.</span><span class="inline-marks">[3]</span></div></li></ol>

In a simple archery game, a player shoots an arrow at a target. The probability of hitting the target depends on the player’s skill and the wind conditions for the day. The wind can either be good or bad, with the probability of a good wind day being \(0.7\).

Dave has a \(70\)% chance of hitting the target on a good wind day and a \(60\)% chance on a bad wind day.

Rafael has an \(80\)% chance of hitting the target on a good wind day and a \(50\)% chance on a bad wind day.

Dave and Rafael decide to compete in a game where each player shoots \(2\) arrows at the target. The player who hits the target more times wins the game.

Find the probability that Dave wins the game on a good wind day.

It is now known that the probability Dave wins the game on a bad wind day is \(0.39\).

Given that Dave wins the game, find the probability that the day was a bad wind day.

2025 TJC P2 Q5

<p>A game at a funfair involves pulling balls of different colours out of a bag. A bag contains \(1\) white ball, \(2\) blue balls, \(3\) green balls and \(4\) red balls. The aim of the game is to pull the white ball from the bag as many times as possible. Players get three attempts. If a red ball is drawn, it is discarded, but any other ball is replaced in the bag.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that a player draws a white ball and two green balls on their three attempts.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that exactly one white ball was drawn, given that a red ball was drawn on the second attempt.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>Events \(A\) and \(B\) are defined as</p>
<p style="margin-left: 14rem">\(A\): A white ball is drawn on the first attempt<br>\(B\): A white ball is drawn on the second attempt</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Determine with reason whether events \(A\) and \(B\) are independent.</span><span class="inline-marks">[2]</span></div></li></ol>

A game at a funfair involves pulling balls of different colours out of a bag. A bag contains \(1\) white ball, \(2\) blue balls, \(3\) green balls and \(4\) red balls. The aim of the game is to pull the white ball from the bag as many times as possible. Players get three attempts. If a red ball is drawn, it is discarded, but any other ball is replaced in the bag.

Find the probability that a player draws a white ball and two green balls on their three attempts.

Find the probability that exactly one white ball was drawn, given that a red ball was drawn on the second attempt.

\(A\): A white ball is drawn on the first attempt
\(B\): A white ball is drawn on the second attempt

Determine with reason whether events \(A\) and \(B\) are independent.

2025 RVHS P2 Q8

<p>A bag contains \(4\) identical red balls and \(6\) identical blue balls.</p>
<ol class="abc-list"><li data-list-item-id="item-a-5" data-list-item-path="item-a-5"><div class="list-item-paragraph">In the first game, the balls are randomly picked by a player from the bag, one at a time, without replacement, until there are no balls left in the bag.</div><ol class="i-list"><li data-list-item-id="item-a-5-i" data-list-item-path="item-a-5/item-a-5-i"><div class="list-item-paragraph">Find the probability that the last ball picked is red.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-a-5-ii" data-list-item-path="item-a-5/item-a-5-ii"><div class="list-item-paragraph">Find the probability that in the first five picks, exactly \(2\) blue balls are picked given that at least \(3\) red balls are picked.<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-b-5" data-list-item-path="item-b-5"><div class="list-item-paragraph">In the second game, each ball picked by a player will be placed back into the bag before the next ball is picked. The player makes a total of \(20\) random picks from the bag and the colour of each pick is recorded.</div><ol class="i-list"><li data-list-item-id="item-b-5-i" data-list-item-path="item-b-5/item-b-5-i"><div class="list-item-paragraph">Find the probability that the colour red is recorded exactly \(4\) times.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-5-ii" data-list-item-path="item-b-5/item-b-5-ii"><div class="list-item-paragraph">Find the probability that the colour red is recorded more than \(4\) times but not more than \(8\) times.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-5-iii" data-list-item-path="item-b-5/item-b-5-iii"><div class="list-item-paragraph"><p>Find the probability that the \(8 ^\mathrm{th}\) pick is the \(6^\mathrm{th}\) time the colour blue is recorded.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-5-iv" data-list-item-path="item-b-5/item-b-5-iv"><div class="list-item-paragraph">The second game is played by \(50\) randomly chosen players. Estimate the probability that the average number of times the colour red is recorded is more than \(8.5\).<span class="inline-marks">[4]</span></div></li></ol></li></ol>

A bag contains \(4\) identical red balls and \(6\) identical blue balls.

In the first game, the balls are randomly picked by a player from the bag, one at a time, without replacement, until there are no balls left in the bag.

In the second game, each ball picked by a player will be placed back into the bag before the next ball is picked. The player makes a total of \(20\) random picks from the bag and the colour of each pick is recorded.

2025 RI P2 Q10

Binomial Distribution

<p>Two fair six-sided dice are thrown. Events \(A\), \(B\) and \(C\) are defined as follows.</p>
<p style="margin-left: 4rem">\(A\): sum of the two scores is odd<br>\(B\): at least one of the two scores is greater than \(4\)<br>\(C\): the two scores are equal</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find, giving your reasons clearly in each case, which pair of the events are</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">mutually exclusive,</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">independent.</span><span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Find \(\mathrm{P}(C \mid B)\).</span><span class="inline-marks">[2]</span></div></li></ol>

Two fair six-sided dice are thrown. Events \(A\), \(B\) and \(C\) are defined as follows.

\(A\): sum of the two scores is odd
\(B\): at least one of the two scores is greater than \(4\)
\(C\): the two scores are equal

Find, giving your reasons clearly in each case, which pair of the events are

2025 NYJC P2 Q6

<p></p>
<p>A bag contains \(4\) black balls and \(6\) white balls. Three balls are drawn from the bag at random, one by one. If a black ball is drawn, it is replaced in the bag. If a white ball is drawn, it is not replaced and \(2\) extra black balls are added to the bag.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that the probability that all balls drawn are white is \(\frac{1}{11}\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that at least one of the balls drawn is black, find the probability that all balls drawn are black.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>A fourth ball is further drawn from the bag at random.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the probability that the third white ball appears on this fourth draw.</span><span class="inline-marks">[3]</span></div></li></ol>

A bag contains \(4\) black balls and \(6\) white balls. Three balls are drawn from the bag at random, one by one. If a black ball is drawn, it is replaced in the bag. If a white ball is drawn, it is not replaced and \(2\) extra black balls are added to the bag.

Show that the probability that all balls drawn are white is \(\frac{1}{11}\).

Given that at least one of the balls drawn is black, find the probability that all balls drawn are black.

A fourth ball is further drawn from the bag at random.

Find the probability that the third white ball appears on this fourth draw.

Math Question Bank