<p>In a game of chance, a player has to spin a fair spinner. The spinner has $7$ sections and an arrow which has an equal chance of coming to rest over any of the $7$ sections. The spinner has $1$ section labelled $\mathbf{R}$, $2$ sections labelled $\mathbf{B}$ and $4$ sections labelled $\mathbf{Y}$ (see diagram).</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/converted/1760955584642-1qpwfpsbvyoi.png" width="400" class="aligncenter" /></figure>
<p>The player then has to throw one of three fair six-sided dice, coloured red, blue or yellow. If the spinner comes to rest over $\mathbf{R}$ the red die is thrown, if the spinner comes to rest over $\mathbf{B}$ the blue die is thrown and if the spinner comes to rest over $\mathbf{Y}$ the yellow die is thrown. The yellow die has one face with $*$ on it, the blue die has two faces with $*$ on it and the red die has three faces with $*$ on it. The player wins the game if the die thrown comes to rest with a face showing $*$ uppermost.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the probability that a player wins a game.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Given that a player wins a game, find the probability that the spinner came to rest over $B$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the probability that a player wins $3$ consecutive games, each time throwing a die of a different colour.</div><p style="text-align: right;">[2]</p></li></ol>

In a game of chance, a player has to spin a fair spinner. The spinner has $7$ sections and an arrow which has an equal chance of coming to rest over any of the $7$ sections. The spinner has $1$ section labelled $\mathbf{R}$, $2$ sections labelled $\mathbf{B}$ and $4$ sections labelled $\mathbf{Y}$ (see diagram).

The player then has to throw one of three fair six-sided dice, coloured red, blue or yellow. If the spinner comes to rest over $\mathbf{R}$ the red die is thrown, if the spinner comes to rest over $\mathbf{B}$ the blue die is thrown and if the spinner comes to rest over $\mathbf{Y}$ the yellow die is thrown. The yellow die has one face with $*$ on it, the blue die has two faces with $*$ on it and the red die has three faces with $*$ on it. The player wins the game if the die thrown comes to rest with a face showing $*$ uppermost.

Find the probability that a player wins a game.

Given that a player wins a game, find the probability that the spinner came to rest over $B$.

Find the probability that a player wins $3$ consecutive games, each time throwing a die of a different colour.

Probability - General

Probability

International Baccalaureate Higher Level Mathematics

IB HL Math

N2016 P2 Q5

(i) Using a probability tree diagram with Spinner (R: $\frac{1}{7}$, B: $\frac{2}{7}$, Y: $\frac{4}{7}$) and Die:

Required Probability $= \frac{1}{7} \times \frac{3}{6} + \frac{2}{7} \times \frac{2}{6} + \frac{4}{7} \times \frac{1}{6}$

(ii) Required Probability $= P(\text{Spinner rest over B} \mid \text{wins game})$

\[= \frac{P(\text{rest over B} \cap \text{wins game})}{P(\text{wins game})}\]

\[= \frac{\frac{2}{7} \times \frac{2}{6}}{\frac{11}{42}}\]

(iii) Required Probability $= P(R^*, B^*, Y^*)$

\[= \left(\frac{1}{7} \times \frac{3}{6}\right)\left(\frac{2}{7} \times \frac{2}{6}\right)\left(\frac{4}{7} \times \frac{1}{6}\right) \times 3!\]

N2016 P2 Q5

N2016 P2 Q5

Video Solution: