Question

A box contains 4 blue, 2 green and 3 red balls and another box contains 4 red, 3 green, and 5 blue balls. A ball is picked up at random from one of the boxes. What is the probability that the ball is blue?

• A. \(\frac{2}{9} \)
• B. \(\frac{5}{24} \)
• C. \(\frac{19}{72} \)
• D. \(\frac{29}{72} \)
• E. \(\frac{31}{72} \)

Answer 1

Answer: (E)

The correct answer is option (E):

\(\frac{31}{72} \)

Here's how to solve this probability problem step-by-step:

First, let's analyze the probabilities associated with each box.

Box 1:

* Total balls: 4 (blue) + 2 (green) + 3 (red) = 9 balls
* Probability of picking a blue ball from Box 1: 4 (blue balls) / 9 (total balls) = 4/9

Box 2:

* Total balls: 4 (red) + 3 (green) + 5 (blue) = 12 balls
* Probability of picking a blue ball from Box 2: 5 (blue balls) / 12 (total balls) = 5/12

Since we're picking a ball from one of the boxes at random, there's a 1/2 chance of selecting either box.

Now, we need to consider the combined probability:

1. Probability of choosing Box 1 AND picking a blue ball: (1/2) * (4/9) = 4/18
2. Probability of choosing Box 2 AND picking a blue ball: (1/2) * (5/12) = 5/24

Finally, to get the overall probability of picking a blue ball, we add the probabilities from each box:

4/18 + 5/24 = 2/9 + 5/24

To add these fractions, we need a common denominator, which is 72.

2/9 = 16/72
5/24 = 15/72

Therefore, 16/72 + 15/72 = 31/72

So, the probability of picking a blue ball is 31/72.