A bag contains a set of ping pong balls numbered from 1 to 12. If you reach in and randomly

pull out 3 of them, what is the expected value of the sum of the numbers on the 3 ping pong balls?
Thanks :)

To find the expected value of the sum of the numbers on the three ping pong balls, we first need to find the probability of each possible outcome.

There are a total of 12C3 = 220 possible ways to select 3 ping pong balls out of 12. Here, 12C3 represents the number of combinations of 12 items taken 3 at a time.

Now, let's calculate the probability of each possible sum:

- If the sum is 3, there is only one way to achieve it: by selecting ping pong balls 1, 1, and 1. So the probability of getting a sum of 3 is 1/220.

- If the sum is 4, there are three different ways to achieve it: (1, 1, 2), (1, 2, 1), and (2, 1, 1). So the probability of getting a sum of 4 is 3/220.

- Continuing this process, we can calculate the probabilities for each possible sum up to 36.

Next, we need to multiply each outcome by its respective probability, and then sum them all up to find the expected value. The formula for the expected value is:

Expected value = (Outcome1 × Probability1) + (Outcome2 × Probability2) + ...

For example, if the sum is 3, the outcome is 3, and the probability is 1/220, so the contribution to the expected value would be (3 × 1/220).

Repeat this calculation for all possible outcomes and their probabilities, and then sum them up to find the expected value.

Keep in mind that performing this calculation manually can be time-consuming, especially when there are numerous possible outcomes. In such cases, using software or programming languages like Python can make the solution much more efficient.