Two coins are taken at random (without replacement) from a bag containing 4 nickels, 8 dimes, and 3 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.)

E(X) = $...

i am fat

To find the expected value E(X), we need to consider all possible outcomes and their corresponding probabilities.

Let's determine the possible outcomes for X and their probabilities.

1. Outcome: Two nickels
The probability of drawing two nickels from the bag is (4/15) * (3/14) = 2/105.
The total value in this case is 5 cents + 5 cents = 10 cents.

2. Outcome: One nickel and one dime
The probability of drawing a nickel and then a dime (in any order) is (4/15) * (8/14) + (8/15) * (4/14) = 32/105.
The total value in this case is 5 cents + 10 cents = 15 cents.

3. Outcome: Two dimes
The probability of drawing two dimes from the bag is (8/15) * (7/14) = 4/15.
The total value in this case is 10 cents + 10 cents = 20 cents.

4. Outcome: One nickel and one quarter
The probability of drawing a nickel and then a quarter (in any order) is (4/15) * (3/14) + (3/15) * (4/14) = 12/105.
The total value in this case is 5 cents + 25 cents = 30 cents.

5. Outcome: One dime and one quarter
The probability of drawing a dime and then a quarter (in any order) is (8/15) * (3/14) + (3/15) * (8/14) = 24/105.
The total value in this case is 10 cents + 25 cents = 35 cents.

6. Outcome: Two quarters
The probability of drawing two quarters from the bag is (3/15) * (2/14) = 1/35.
The total value in this case is 25 cents + 25 cents = 50 cents.

Now, we calculate the expected value E(X):

E(X) = (10 * 2/105) + (15 * 32/105) + (20 * 4/15) + (30 * 12/105) + (35 * 24/105) + (50 * 1/35)

E(X) = 0.1905 + 0.4571 + 2.6667 + 0.8571 + 0.8000 + 1.4286
E(X) = 6.3990

Therefore, E(X) is approximately $6.3990.

To find the expected value (E) of the random variable X, we need to compute the sum of the products between each possible value of X and its corresponding probability.

First, let's find all the possible values of X and their corresponding probabilities.

There are a total of 15 coins in the bag:
- 4 nickels, which have a value of 5 cents each.
- 8 dimes, which have a value of 10 cents each.
- 3 quarters, which have a value of 25 cents each.

To find the possible values of X, we need to consider all the possible ways of selecting two coins from the bag.

Let's consider the value of the first coin selected and the value of the second coin selected, and compute the total value X.

1. If we select two nickels: the value of the first coin is 5 cents, and the value of the second coin is also 5 cents. So, the total value is 5 + 5 = 10 cents. The probability of this happening is (4/15) * (3/14) = 12/210.

2. If we select a nickel and a dime: the value of the first coin is 5 cents, and the value of the second coin is 10 cents. So, the total value is 5 + 10 = 15 cents. The probability of this happening is (4/15) * (8/14) = 32/210.

3. If we select a nickel and a quarter: the value of the first coin is 5 cents, and the value of the second coin is 25 cents. So, the total value is 5 + 25 = 30 cents. The probability of this happening is (4/15) * (3/14) = 12/210.

4. If we select two dimes: the value of the first coin is 10 cents, and the value of the second coin is also 10 cents. So, the total value is 10 + 10 = 20 cents. The probability of this happening is (8/15) * (7/14) = 56/210.

5. If we select a dime and a nickel: the value of the first coin is 10 cents, and the value of the second coin is 5 cents. So, the total value is 10 + 5 = 15 cents. The probability of this happening is (8/15) * (4/14) = 32/210.

6. If we select a dime and a quarter: the value of the first coin is 10 cents, and the value of the second coin is 25 cents. So, the total value is 10 + 25 = 35 cents. The probability of this happening is (8/15) * (3/14) = 24/210.

7. If we select a quarter and a nickel: the value of the first coin is 25 cents, and the value of the second coin is 5 cents. So, the total value is 25 + 5 = 30 cents. The probability of this happening is (3/15) * (4/14) = 12/210.

8. If we select a quarter and a dime: the value of the first coin is 25 cents, and the value of the second coin is 10 cents. So, the total value is 25 + 10 = 35 cents. The probability of this happening is (3/15) * (8/14) = 24/210.

9. If we select two quarters: the value of the first coin is 25 cents, and the value of the second coin is also 25 cents. So, the total value is 25 + 25 = 50 cents. The probability of this happening is (3/15) * (2/14) = 6/210.

Now, let's compute the expected value by summing the products of each possible value of X and its probability:

E(X) = (10 * (12/210)) + (15 * (32/210)) + (30 * (12/210)) + (20 * (56/210)) + (15 * (32/210)) + (35 * (24/210)) + (30 * (12/210)) + (35 * (24/210)) + (50 * (6/210))

E(X) ≈ 18.7619

Therefore, E(X) ≈ $18.7619.

.2611