Sara tossed a fair coin five times, and Kaleb tossed a fair coin three times. There were five heads and three tails in the eight tosses. What is the probability that either Sara or Kaleb tossed exactly three heads? Express your answer as a common fraction.

that is wrong

5/7

To find the probability that either Sara or Kaleb tossed exactly three heads, we need to calculate the probability for each person and then add them together.

First, let's consider Sara. From the given information, we know that Sara tossed the coin five times and there were a total of five heads and three tails.

To count the number of ways she could have exactly three heads, we can use the binomial coefficient, also known as "n choose k." In this case, the formula is:

C(n, k) = n! / (k!(n - k)!)

where n represents the total number of tosses and k represents the number of successful outcomes.

For Sara, we have n = 5 (five coin tosses) and k = 3 (three heads). So, the number of ways Sara can get exactly three heads is:

C(5, 3) = 5! / (3!(5 - 3)!) = 5! / (3! * 2!) = (5 * 4 * 3!) / (3! * 2!) = (5 * 4) / 2 = 10.

The total number of possible outcomes for Sara is 2^5 since each coin toss has two possible outcomes (heads or tails). So, the probability that Sara tossed exactly three heads is 10/2^5.

Now, let's do the same calculations for Kaleb. From the given information, we know that Kaleb tossed the coin three times and there were a total of five heads and three tails in all the tosses.

Using the same binomial coefficient formula, we have n = 3 (three coin tosses) and k = 3 (three heads). So, the number of ways Kaleb can get exactly three heads is:

C(3, 3) = 3! / (3!(3 - 3)!) = 3! / (3! * 0!) = 1.

The total number of possible outcomes for Kaleb is 2^3 since each coin toss has two possible outcomes (heads or tails). So, the probability that Kaleb tossed exactly three heads is 1/2^3.

Now, to find the probability that either Sara or Kaleb tossed exactly three heads, we add their individual probabilities together:

(10/2^5) + (1/2^3) = 10/32 + 1/8 = 10/32 + 4/32 = 14/32.

Simplifying the fraction, we get:

14/32 = 7/16.

Therefore, the probability that either Sara or Kaleb tossed exactly three heads is 7/16.

Probability that Sarah throws exactly 3 heads:

Binomial[5,3] 2^(-5) = 10/32 = 5/16

Probability that Sarah does not throw excatly 3 heads is:

1 - 5/16 = 11/16

Probability that Kaleb throws exactly 3 heads:

1/8

Probability that Kaleb does not throw excatly 3 heads is:

1-1/8 = 7/8

Probability that neither Kaleb nor Sarah throw exactly 3 heads is:

11/16 * 7/8 = 77/128

Probability that either Kaleb or Sarah throw exactly 3 heads is:

1 - 77/128 = 51/128