Two coins are taken at random (without replacement) from a bag containing 7 nickels, 5 dimes, and 6 quarters. Let X denote the random variable given by the total value of the two coins. Find E(X). (Round your answer to four decimal places.)
1
To find the expected value (E(X)) of a random variable, we need to multiply each possible outcome by its corresponding probability and sum them up.
Let's consider all the possible outcomes and their probabilities:
1. Nickels - If we choose two nickels.
Probability = (7/18) * (6/17) = 42/306
Total value = 5 cents + 5 cents = 10 cents
2. Dimes - If we choose two dimes.
Probability = (5/18) * (4/17) = 20/306
Total value = 10 cents + 10 cents = 20 cents
3. Quarters - If we choose two quarters.
Probability = (6/18) * (5/17) = 30/306
Total value = 25 cents + 25 cents = 50 cents
4. Mixed coins - If we choose one nickel and one dime.
Probability = (7/18) * (5/17) + (5/18) * (7/17) = 70/306
Total value = 5 cents + 10 cents = 15 cents
5. Mixed coins - If we choose one nickel and one quarter.
Probability = (7/18) * (6/17) + (6/18) * (7/17) = 84/306
Total value = 5 cents + 25 cents = 30 cents
6. Mixed coins - If we choose one dime and one quarter.
Probability = (5/18) * (6/17) + (6/18) * (5/17) = 60/306
Total value = 10 cents + 25 cents = 35 cents
Now, we can calculate the expected value:
E(X) = (10 * 42/306) + (20 * 20/306) + (50 * 30/306) + (15 * 70/306) + (30 * 84/306) + (35 * 60/306)
E(X) = 420/306 + 400/306 + 1500/306 + 1050/306 + 2520/306 + 2100/306
E(X) = 6590/306
E(X) ≈ 21.52 (rounded to four decimal places)
Therefore, the expected value of the total value of the two coins, E(X), is approximately 21.52.