Olivia tossed two fair two-sided coins. In a game, she earns 4 points if exactly one coin lands on heads and 6 points if both coins land on heads. If points are not earned for tails, what is the expected value of the points earned?
A. 3 1/3
B. 3 1/2 ***
C. 5
D. 6
The answers to the LEAP Math Midtest- 7 (GT) are:
C
B
D
B
A
A
C
D
B
D
A
C
B
C
A
D
C
D
B
A
C
A
B
D
C
A
D
B
C
C
A
B
D
C
D
B
C
D
B
A
C
B
D
A
B
A
C
D
D
B
coin 1 **h t
coin 2 h 6 4
coin 2 t 4 0
p(6 points) = 1/4
p(4 points) = 1/2
p(0 points) = 1/4
(1/4)6 + (1/2)4 = 2 + 1.5 = 3.5
so I agree with you
To find the expected value of the points earned, we need to calculate the weighted average of the values we can earn (4 points and 6 points) based on their probabilities.
Let's calculate the probabilities first:
There are 2 possibilities when tossing two coins: both coins can land on heads (HH) or only one coin can land on heads (either HT or TH).
For the HH case: The probability of the first coin landing on heads is 1/2, and the probability of the second coin also landing on heads is 1/2. Therefore, the probability of both coins landing on heads is (1/2) * (1/2) = 1/4.
For the HT/TH case: Since there are two possible ways for one coin to land on heads (HT or TH), the probability of one coin landing on heads is (1/2) + (1/2) = 1.
Now, let's calculate the expected value:
The expected value is the sum of the products of each value and its corresponding probability.
Expected value = (4 points * probability of HT/TH) + (6 points * probability of HH)
= (4 * 1) + (6 * 1/4)
= 4 + 1.5
= 5.5
Therefore, the expected value of the points earned is 5.5, which is closer to option C (5) than any other option. However, none of the answer choices match the calculated value exactly.
So, the closest answer choice to the calculated expected value of 5.5 is option B (3 1/2).