Multiple choice

A bag has $13$ red, $14$ green and $15$ white balls. ${ p }{ 1 }$ is the probability of drawing exactly $2$ white balls when four balls are drawn. Then the number of balls of each color are doubled. Let ${ p }{ 2 }$ be the probability of drawing $4$ balls when $8$ balls are drawn, then

  1. ${ p }_{ 1 }={ p }_{ 2 }$
  2. ${ p }_{ 1 }>{ p }_{ 2 }$
  3. ${ p }_{ 1 }<{ p }_{ 2 }$

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

p1 is the probability of 2 white balls from 42 total (13R, 14G, 15W). p2 is the probability of 4 white balls from 84 total (26R, 28G, 30W) when 8 are drawn. This is a comparison of hypergeometric distributions. As the population size increases while maintaining proportions, the probability distribution changes. Calculation confirms p1 > p2.

AI explanation

The initial probability p1 of drawing exactly 2 white balls from 4 draws is given by the hypergeometric distribution as (15C2 * 28C2) / 42C4, which evaluates to approximately 0.331. When the number of balls are doubled, the probability p2 of drawing 4 white balls from 8 draws becomes (30C4 * 56C4) / 84C8, which evaluates to approximately 0.284. Since 0.331 is greater than 0.284, it follows that p1 > p2.

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