<p>Two fair dice, one red and the other green, are thrown.</p>
<p style="text-align: center">$A$ is the event: The score on the red die is divisible by $3$.</p>
<p style="text-align: center">$B$ is the event: The sum of the two scores is $9$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Justifying your conclusion, determine whether $A$ and $B$ are independent.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">The experiment is repeated multiple times. How many times must the experiment be conducted, so that the probability that $B$ occurs at least once exceeds $0.98$?</span></div><p style="text-align: right;">[3]</p></li></ol>

Two fair dice, one red and the other green, are thrown.

$A$ is the event: The score on the red die is divisible by $3$.

$B$ is the event: The sum of the two scores is $9$.

Justifying your conclusion, determine whether $A$ and $B$ are independent.

The experiment is repeated multiple times. How many times must the experiment be conducted, so that the probability that $B$ occurs at least once exceeds $0.98$?

Probability - General

Probability

International Baccalaureate Higher Level Mathematics

IB HL Math

N2006 P2 Q23 [Modified]

$P(A) = \frac{2}{6} = \frac{1}{3}$ (red die shows 3 or 6)

$P(B) = \frac{4}{36} = \frac{1}{9}$ (pairs: (3,6), (4,5), (5,4), (6,3))

$P(A \cap B) = \frac{2}{36} = \frac{1}{18}$ (pairs: (3,6), (6,3))

\[P(A) \times P(B) = \frac{1}{3} \times \frac{1}{9} = \frac{1}{27} \ne P(A \cap B) = \frac{1}{18}\]

$\therefore A$ and $B$ are not independent events.

\[P(B \text{ occurs at least once}) = 1 - P(B \text{ never occurs}) = 1 - \left(\frac{8}{9}\right)^n\]

\[1 - \left(\frac{8}{9}\right)^n > 0.98\]

Since $\ln\frac{8}{9} < 0$, dividing flips inequality:

\[n > \frac{\ln 0.02}{\ln \frac{8}{9}} = 33.21\]

$\therefore$ Least $n = 34$. The experiment must be conducted at least 34 times.

N2006 P2 Q23 [Modified]

N2006 P2 Q23 [Modified]

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