Christine Wong has asked Dave and Mike to help her move into a new apartment on Sunday morning. She has asked them both in case one of them does not show up. From past experience, Christine knows that there is a 55% chance that Dave will not show up and a 45% chance that Mike will not show up. Dave and Mike do not know each other and their decisions can be assumed to be independent.a. What is the probability that both Dave and Mike will show up? (Round your answer to 2 decimal places.)b. What is the probability that at least one of them will show up?c. What is the probability that neither Dave nor Mike will show up? (Round your answer to 2 decimal places.)

Two events are independent if the incidence of one of them doesn't affect the probability of the other. Let A and B be two events, if they are independent:

[tex]P(A\cap B)=P(A)\times P(B)[/tex]

It's called the Rule of Product

Christine knows that there is a 55% (0.55) chance that Dave will not show up and a 45% (0.45) chance that Mike will not show up. Thus

[tex]P(A)=0.55,\ P(B)=0.45[/tex]

To completely solve this problem, we need to know the probability of both negated events, that is, the probability that Dave or Mike actually show up. This is written as

[tex]P(A')=1-0.55=0.45\\\ P(B')=1-0.45=0.55[/tex]

a. What is the probability that both Dave and Mike will show up?

The required probability is the product of both negated events, thus

[tex]P=0.45\times 0.55=0.25[/tex]

b. What is the probability that at least one of them will show up?

It means that we must assume that A happens, B happens, or both events happen. It's easier to find the probability that both events won't happen and find its negation

[tex]P(A\ and\ B)=0.55\times 0.45=0.25[/tex]

The required probability is

[tex]P=1-0.25=0.75[/tex]

c. What is the probability that neither Dave nor Mike will show up?

We have already found that value

[tex]P(A and B)=0.55\times 0.45=0.25[/tex]