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I have $5$ marbles numbered $1$ through $5$ in a bag. Suppose i take out two different marbles at random. What is the expected value of the sum of the numbers on the marbles?

User Clarissa
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3 votes

Answer:

Explanation:

To find the expected value of the sum of the numbers on the two marbles, we need to consider all possible outcomes and their respective probabilities.

There are a total of 5 marbles, and we are drawing 2 different marbles at random. The sum of the numbers on the marbles can range from 3 (1+2) to 9 (4+5).

To calculate the expected value, we need to find the probability of each possible sum and multiply it by the corresponding sum. Let's calculate it step by step:

Sum 3: There is only one way to get a sum of 3, which is by drawing marbles numbered 1 and 2. The probability of this outcome is (1/5) * (1/4) = 1/20.

Sum 4: There are two ways to get a sum of 4: (1,3) or (2,2). The probability of each outcome is (1/5) * (2/4) = 1/10.

Sum 5: There are two ways to get a sum of 5: (1,4) or (2,3). The probability of each outcome is (1/5) * (2/4) = 1/10.

Sum 6: There are two ways to get a sum of 6: (1,5) or (2,4). The probability of each outcome is (1/5) * (2/4) = 1/10.

Sum 7: There are two ways to get a sum of 7: (3,4) or (2,5). The probability of each outcome is (1/5) * (2/4) = 1/10.

Sum 8: There is only one way to get a sum of 8, which is by drawing marbles numbered 4 and 4. The probability of this outcome is (1/5) * (1/4) = 1/20.

Sum 9: There is only one way to get a sum of 9, which is by drawing marbles numbered 4 and 5. The probability of this outcome is (1/5) * (1/4) = 1/20.

Now we can calculate the expected value by multiplying each sum by its corresponding probability and summing them up:

(3 * 1/20) + (4 * 1/10) + (5 * 1/10) + (6 * 1/10) + (7 * 1/10) + (8 * 1/20) + (9 * 1/20) = 6/20 + 4/10 + 5/10 + 6/10 + 7/10 + 4/20 + 9/20 = 49/20 = 2.45

Therefore, the expected value of the sum of the numbers on the marbles is 2.45.

hope this helped!

User Sam Yi
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