Two dice are rolled and two fair coins are tossed. Let X be the sum of the number of spots that show on the top faces of the dice and the number of coins that land heads up. The expected value of X is _____ .
Well, the die are independent.
expected dots from one die:1/6(1+2+3+4+5+6)=21/6
expected sum:= 42/6=7
expected number of heads: 1/2 * 2
expected value of X is 8
A lottery offers one $900 prize, one $600 prize, three $400 prizes, and five $100 prizes. One thousand tickets are sold at $6 each. Find the expectation of a person buys five tickets.
To find the expected value of X, we need to determine all possible outcomes and their associated probabilities.
Let's start by listing all possible outcomes for rolling two dice and tossing two coins:
- Dice outcomes: (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)
- Coin outcomes: (H, H), (H, T), (T, H), (T, T)
Now let's calculate the sum of spots on the dice and the number of heads:
- (1, 1): Sum = 1 + 1 = 2, Number of heads = 2
- (1, 2): Sum = 1 + 2 = 3, Number of heads = 1
- (1, 3): Sum = 1 + 3 = 4, Number of heads = 1
- ...
- (6, 6): Sum = 6 + 6 = 12, Number of heads = 0
Next, we need to calculate the probability of each outcome:
- Probability of rolling a specific outcome on two dice: 1/36 (since there are 36 equally likely outcomes for rolling two dice)
- Probability of getting a specific outcome on two coins: 1/4 (since there are 4 equally likely outcomes for tossing two coins)
Now we can determine the expected value of X by multiplying each outcome by its probability and summing them up:
Expected value of X = (2 * (1/36 * 1/4)) + (3 * (1/36 * 1/2)) + (4 * (1/36 * 1/2)) + ... + (12 * (1/36 * 1/2))
Simplifying this expression will give us the expected value of X.