Question

there are two boxes. the first one has 5 red and 5 green balls. the second box has 5 red and 2 green balls. you randomly choose a box and then select one ball from the box. what is the probability that you selected first box if the ball you picked is red

Answer 1

The probability that the red ball was selected from the first box given a red ball was picked is 7/17. This is calculated using Bayes' theorem.

Step-by-step explanation:

The question seeks the probability that a red ball was picked from the first box, given that a red ball was actually picked. The problem uses Bayes' theorem to find the posterior probability.

Step-by-Step Solution:

Let's denote the event of picking the first box as B1 and the second box as B2. The event of picking a red ball will be denoted as R.

1. Calculate the prior probabilities: P(B1) = 1/2 (since there are two boxes) and P(B2) = 1/2.
2. Calculate the likelihoods: P(R|B1) = 5/10 (because there are 5 red balls out of 10 in the first box) and P(R|B2) = 5/7 (because there are 5 red balls out of 7 in the second box).
3. Use Bayes' theorem to find P(B1|R) = [P(B1) * P(R|B1)] / [P(B1) * P(R|B1) + P(B2) * P(R|B2)].
4. Perform the calculation: P(B1|R) = [(1/2) * (5/10)] / [(1/2) * (5/10) + (1/2) * (5/7)] = 7/17.

Therefore, the probability that the first box was chosen given that the selected ball was red is 7/17.