A fair coin is tossed three times. What is the probability that exactly two heads occur, given that a. the first outcome was a tail b. the first two outcomes were heads c. the first two outcomes were tails

a) So 25% probability that exactly two heads occur, given that the first outcome was a tail.

b) 50% probability that exactly two heads occur, given that the first two outcomes were heads.

c) 0% probability that exactly two heads occur, given that the first two outcomes were tails.

Step-by-step explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

We have the following sample space, that is, the possible outcomes:

In which h is heads and t is tails

h - h - h

h - h - t

h - t - h

h - t - t

t - h - h

t - h - t

t - t - h

t - t - t

What is the probability that exactly two heads occur, given that

a. the first outcome was a tail

Four outcomes in which the first outcome was a tail. They are:

t - h - h

t - h - t

t - t - h

t - t - t

In only 1 one them, exactly two heads occur.

1/4 = 0.25

So 25% probability that exactly two heads occur, given that the first outcome was a tail.

b. the first two outcomes were heads

Two possibilities in which the first two outcomes were heads.

h - h - h

h - h - t

In 1 of them, we have exactly two heads.

1/2 = 0.5

So 50% probability that exactly two heads occur, given that the first two outcomes were heads.

c. the first two outcomes were tails

Two possibilities in which the first two outcomes were tails.

t - t - h

t - t - t

In none of them we have exactly 2 heads.

0% probability that exactly two heads occur, given that the first two outcomes were tails.