First slide

Random variable and probability distribution

Question

A box contains 24 identical balls of which 12 are white and 12 are black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the 4th time on the 7th draw is

Moderate

A

5/64
B

27/32
C

5/32
D

1/2

Solution

In any trial P(getting white ball) = p(W) = 1/2
P(getting black ball) = P(B) = 1/2
Required event will occur if in the first six trial, 3 white balls are drawn in any of the 3 trails from six.
In remaining three trails, black must be drawn.
Three trials in which black balls is drawn can be selected in $^{6} C_{3} = 20$ ways.
$∴$ Required probability = P(WWWBBB in any order) x P(W)
$= 20 \times {(\frac{1}{2})}^{3} \times {(\frac{1}{2})}^{3} \times \frac{1}{2} = \frac{5}{32}$

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