Probability--

There are two boxes...the first box has 3 black balls and 1 blue ball in it.

The second box has 6 black balls and 2 in blue balls in it.

a. Which of the following two boxes would you choose if you were picking 1 ball and wanted black? Justify your answer.

b. Then what if you were picking 2 balls (without replacement) and wanted 2 black ones?

prob of getting one black from 1st box

= 3/4
prob of getting one black from 2nd box
= 6/8 = 3/4

mmmmmhhhh?

prob(2 blacks from 1st) = (3/4)(2/3) = 1/2
prob(2 blacks from 2nd) = (6/8)(5/7) = 15/28

mmmh again!

To determine which box to choose if you want to pick a black ball, we need to compare the probabilities of drawing a black ball from each box.

a. For the first box:
- Total number of balls: 3 black + 1 blue = 4 balls
- Probability of drawing a black ball = Number of black balls / Total number of balls = 3/4 = 0.75

For the second box:
- Total number of balls: 6 black + 2 blue = 8 balls
- Probability of drawing a black ball = Number of black balls / Total number of balls = 6/8 = 0.75

Both boxes have the same probability (0.75) of drawing a black ball. Therefore, if you were picking 1 ball and wanted black, it does not matter which box you choose.

b. When picking 2 balls (without replacement) and wanting 2 black ones, we need to consider the probabilities separately for each box.

For the first box:
- Probability of drawing the first black ball = Number of black balls / Total number of balls = 3/4 = 0.75
- After drawing the first black ball, the total number of balls becomes 3 (2 black + 1 blue). Therefore, the probability of drawing a second black ball is now 2/3 ≈ 0.67.

For the second box:
- Probability of drawing the first black ball = Number of black balls / Total number of balls = 6/8 = 0.75
- After drawing the first black ball, the total number of balls becomes 7 (5 black + 2 blue). Therefore, the probability of drawing a second black ball is now 5/7 ≈ 0.71.

Comparing the probabilities:
- For the first box, the probability of drawing 2 black balls is 0.75 * 0.67 ≈ 0.5025.
- For the second box, the probability of drawing 2 black balls is 0.75 * 0.71 ≈ 0.5325.

Therefore, if you were picking 2 balls (without replacement) and wanted 2 black ones, the second box would have a slightly higher probability of fulfilling that criteria.