In a game you flip a coin twice, and record the number of heads that occur. You get 10 points for 2 heads, zero points for 1 head, and 5 points for no heads. What is the expected value for the number of points you’ll win per turn?
3.75
To calculate the expected value, we need to multiply each possible outcome by its corresponding probability and sum them up.
Let's consider the possible outcomes:
1. Two heads: This occurs with a probability of (1/2 * 1/2) = 1/4. In this case, you receive 10 points.
2. One head: This occurs with a probability of (1/2 * 1/2) = 1/4. In this case, you receive zero points.
3. No heads: This occurs with a probability of (1/2 * 1/2) = 1/4. In this case, you receive 5 points.
Now let's calculate the expected value:
Expected Value = (Probability of Two heads * Points for Two heads) + (Probability of One head * Points for One head) + (Probability of No heads * Points for No heads)
Expected Value = (1/4 * 10) + (1/4 * 0) + (1/4 * 5)
Expected Value = 10/4 + 0/4 + 5/4
Expected Value = 15/4
Expected Value = 3.75
Therefore, the expected value for the number of points you'll win per turn is 3.75.
To calculate the expected value, you need to multiply each outcome by its corresponding probability and then sum them up. Let's break it down step by step:
1. Determine the possible outcomes:
In this game, there are three possible outcomes based on the number of heads obtained:
- Outcome 1: Getting 2 heads (HH)
- Outcome 2: Getting 1 head (HT or TH)
- Outcome 3: Getting no heads (TT)
2. Assign probabilities to each outcome:
The probability of getting heads (H) in a coin flip is 0.5, and the probability of getting tails (T) is also 0.5.
- Probability of Outcome 1: P(H) * P(H) = 0.5 * 0.5 = 0.25
- Probability of Outcome 2: P(H) * P(T) + P(T) * P(H) = 0.5 * 0.5 + 0.5 * 0.5 = 0.5
- Probability of Outcome 3: P(T) * P(T) = 0.5 * 0.5 = 0.25
3. Determine the points associated with each outcome:
- Outcome 1: Getting 2 heads = 10 points
- Outcome 2: Getting 1 head = 0 points
- Outcome 3: Getting no heads = 5 points
4. Calculate the expected value:
Multiply each outcome by its probability and sum them up:
Expected Value = (Outcome 1 * Probability of Outcome 1) + (Outcome 2 * Probability of Outcome 2) + (Outcome 3 * Probability of Outcome 3)
Expected Value = (10 * 0.25) + (0 * 0.5) + (5 * 0.25)
Expected Value = 2.5 + 0 + 1.25
Expected Value = 3.75
Therefore, the expected value for the number of points you'll win per turn in this game is 3.75.
Outcomes:
HH
HT
TH
TT
expected value = (1/4)(10) + (2/4)(1) + (1/4)(5)
= 17/4 = 4.25