<p>A bag contains $3$ red balls and $3$ green balls. Balls are drawn from the bag at random, one by one and without replacement.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that the probability that the first $3$ balls drawn are red is $\frac{1}{20}$.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that the first $3$ balls drawn consist of $2$ red balls and $1$ green ball (in an order). Hence, or otherwise show that the probability that the third red appear on the fourth draw is $\frac{3}{20}$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that the third red ball appears on the fifth draw.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">The random variable $X$ is the number of draws required up to and including the one on which the third red ball appears. Tabulate the probability distribution of $X$ and find $\mathrm{E}\left( X \right)$.</span></div><p style="text-align: right;">[4]</p></li></ol>

A bag contains $3$ red balls and $3$ green balls. Balls are drawn from the bag at random, one by one and without replacement.

Show that the probability that the first $3$ balls drawn are red is $\frac{1}{20}$.

Find the probability that the first $3$ balls drawn consist of $2$ red balls and $1$ green ball (in an order). Hence, or otherwise show that the probability that the third red appear on the fourth draw is $\frac{3}{20}$.

Find the probability that the third red ball appears on the fifth draw.

The random variable $X$ is the number of draws required up to and including the one on which the third red ball appears. Tabulate the probability distribution of $X$ and find $\mathrm{E}\left( X \right)$.

Probability - General

Probability

International Baccalaureate Higher Level Mathematics

IB HL Math

N1995 P2 Q6

Required Probability

$=\mathrm{P}(\text{first 3 are red})$

$=\dfrac{{}^{3}C_{3}}{{}^{6}C_{3}}$ or $\dfrac{3}{6}\times\dfrac{2}{5}\times\dfrac{1}{4}$

$=\dfrac{1}{20}$ (shown)

Required Probability

$=\mathrm{P}(\text{first 3 consist of 2R, 1G})$

$=\dfrac{{}^{3}C_{2}\times{}^{3}C_{1}}{{}^{6}C_{3}}$ or $\dfrac{3}{6}\times\dfrac{2}{5}\times\dfrac{3}{4}\times\dfrac{3!}{2!}$

$=\dfrac{9}{20}$

$\mathrm{P}(\text{3rd red appears on 4th draw})$

$=\mathrm{P}(\text{first 3 consist of 2R, 1G})\times\mathrm{P}(\text{4th is red})$

$=\dfrac{9}{20}\times\dfrac{1}{3}$

$=\dfrac{3}{20}$ (shown)

$\mathrm{P}(\text{3rd red on 5th draw})$

$=\mathrm{P}(\text{2R, 2G in first 4})\times\mathrm{P}(\text{5th is red})$

$=\left[\dfrac{3}{6}\times\dfrac{2}{5}\times\dfrac{3}{4}\times\dfrac{2}{3}\times\dfrac{4!}{2!\,2!}\right]\times\dfrac{1}{2}$

$=\dfrac{3}{10}$

| $x$ | 3 | 4 | 5 | 6 |
|---|---|---|---|---|
| $\mathrm{P}(X=x)$ | $\dfrac{1}{20}$ | $\dfrac{3}{20}$ | $\dfrac{3}{10}$ | $\dfrac{1}{2}$ |

$\mathrm{P}(X=6)=1-\dfrac{1}{20}-\dfrac{3}{20}-\dfrac{3}{10}=\dfrac{1}{2}$

$\mathrm{E}(X)=3\times\dfrac{1}{20}+4\times\dfrac{3}{20}+5\times\dfrac{3}{10}+6\times\dfrac{1}{2}$

$=\dfrac{3}{20}+\dfrac{12}{20}+\dfrac{30}{20}+\dfrac{60}{20}$

$=\dfrac{105}{20}=5\dfrac{1}{4}$

N1995 P2 Q6

N1995 P2 Q6

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