GATE 2022 IN Q.32: solution and answer

GATE 2022 · INQ.32 · 1 mark · NAT

A bag contains six red balls and four blue balls. If three balls are drawn in succession without replacement, the probability that the second and third balls drawn are red is ________ (round off to two decimal places)

See the question in the paper

Tap to open full size

Answer and solution

ANSWER

The probability that the second and third balls drawn are both red is $\frac{1}{3} \approx 0.33$ .

Official answer: 0.32 to 0.34Source: Official answer key, GATE 2022 IN (IIT Kharagpur)

Let $R_{k}$ and $B_{k}$ mean that the $k$ -th ball drawn is red or blue. The question does not fix the colour of the first ball, so split on it with the total probability theorem: $P (R_{2} R_{3}) = P (R_{1}) P (R_{2} R_{3} ∣ R_{1}) + P (B_{1}) P (R_{2} R_{3} ∣ B_{1})$ where $P (R_{1}) = \frac{6}{10}$ and $P (B_{1}) = \frac{4}{10}$ .
If the first ball is red, 9 balls remain and 5 of them are red: $P (R_{2} R_{3} ∣ R_{1}) = \frac{5}{9} \cdot \frac{4}{8} = \frac{20}{72}$ .
If the first ball is blue, 9 balls remain and all 6 red balls are still there: $P (R_{2} R_{3} ∣ B_{1}) = \frac{6}{9} \cdot \frac{5}{8} = \frac{30}{72}$ .
Add the two cases: $P (R_{2} R_{3}) = \frac{6}{10} \cdot \frac{20}{72} + \frac{4}{10} \cdot \frac{30}{72} = \frac{120 + 120}{720} = \frac{240}{720} = \frac{1}{3}$
Check by symmetry. When balls are drawn without replacement, any two draw positions have the same joint distribution, so $P (R_{2} R_{3}) = P (R_{1} R_{2}) = \frac{6}{10} \cdot \frac{5}{9} = \frac{30}{90} = \frac{1}{3}$ . Both methods give the same value.
$\frac{1}{3} = 0.333 \dots \approx 0.33$ .

KEY IDEA

The first ball's colour is unknown, so condition on it with the total probability theorem. There is also a shortcut: without replacement, every draw position is symmetric, so the chance that draws 2 and 3 are red equals the chance that draws 1 and 2 are red, which is $\frac{6}{10} \cdot \frac{5}{9} = \frac{1}{3}$ .

WATCH OUT FOR

Taking the first ball to be blue, or to be red, and computing only that case. This gives $\frac{6}{9} \cdot \frac{5}{8} \approx 0.42$ or $\frac{5}{9} \cdot \frac{4}{8} \approx 0.28$ . Another mistake is computing the probability of the full sequence (a colour for draw 1, then red, red) and forgetting to add the two cases.

From the GATE 2022 IN paper. Every question of this paper · Download the full paper · All 33 Probability & statistics questions, with notes

GATE 2022 IN Q.32: solution and answer

Answer and solution

More Probability & statistics questions