Multiple choice

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 4$^{th}$ time on the 7$^{th}$ draw is

  1. $\displaystyle \frac{5}{64}$
  2. $\displaystyle \frac{27}{32}$
  3. $\displaystyle \frac{5}{32}$
  4. $\displaystyle \frac{1}{2}$
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C Correct answer
Explanation

For the 4th white ball to appear on the 7th draw, there must be exactly 3 white balls in the first 6 draws, and the 7th draw must be white. The probability of white is 1/2. The number of ways to choose 3 positions for white in the first 6 draws is 6C3 = 20. The probability is 20 * (1/2)^6 * (1/2) = 20 / 128 = 5/32.

AI explanation

Using the negative binomial distribution, we need exactly 3 white balls drawn in the first 6 draws, followed by a white ball on the 7th draw. The probability of 3 white balls in 6 draws is given by 6C3 multiplied by (1/2)^3 multiplied by (1/2)^3, which equals 20 divided by 64. Multiplying this by the 1/2 probability of drawing a white ball on the 7th draw yields 20 divided by 128, which simplifies to 5/32.

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