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 53% chance that Dave will not show up and a 43% 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? (Round your answer to 2 decimal places.) c. What is the probability that neither Dave nor Mike will show up? (Round your answer to 2 decimal places.)

Dave shows up but Mike does not. The probability of Dave to show up is (1-0.53)=0.47 and the probability that Mike does not is 0.43. Then, as these events are independent, the probability for both of them to happen is (0.47)(0.43)=0.2021=20.21%
Mike shows up but Dave does not. The probability of Mike to show up is (1-0.43)=0.57 and the probability that Mike does not is 0.53. Then, as these events are independent, the probability for both of them to happen is (0.57)(0.53)=0.3021=30.21%

Therefore, the probability that at least one of them will show up? is 20.21%+30.21%=50.42%.

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

As decisions of showing up can be assumed to be independent, the probability will be given by

[tex](0.53)(0.43)=0.2279[/tex]=22.79%

astha8579 astha8579 · Answer 2 · 2022-02-25T06:28:49+01:00

Probability of an event is the measure of its chance of occurrence. The probabilities for Dave and Mike showing up are:

P(Dave and Mike both will show up) = 0.2679 = 26.79%
P(At least one of them will show up) = 0.7721 = 77.21%

How to convert percent to probability?

Percent counts the number compared to 100 whereas probability counts it compare to 1.

So, if we have a%, that means for each 100, there are 'a' parts. If we divide each of them with 100, we get:

For each 1, there are a/100 parts.

Thus, 50% = 50/100 = 0.50 (in probability)

What is the addition rule of probability for two events?

For two events A and B, we have:

Probability that event A or B occurs = Probability that event A occurs + Probability that event B occurs - Probability that both the event A and B occur simultaneously.

This can be written symbolically as:

[tex]P(A \cup B) = P(A) + P(B) - P(A \cap B)[/tex]

Let for the given condition, the events be

A = Dave will show up
A' = Dave won't show up (complement event of A)
B = Mike will show up
B' = Mike won't show up (complement event of event B)

Then, by the given data, we have the probabilities as:

P(A') = 53% = 0.53
P(A) = 1-P(A) = 1-0.53 = 0.47
P(B') = 43% = 0.43
P(B) = 1 - 0.43 = 0.57

The needed probabilities are calculated as:

Case 1: P(Dave and Mike both will show up)

P(Dave and Mike both will show up) = P(A ∩ B)

Since it is given that A' and B' (and thus A and B) are independent of each other.
This means

P(A ∩ B) = P(A)P(B) (by definition of independent events)

Thus, P(A ∩ B) = 0.47 × 0.57 = 0.2679

Case 2: P(At least one of them will show up)

P(At least one of them will show up) = P(A∪B)

Using the addition rule of probability for two events, we get:

P(A∪B) = P(A) - P(B) - P(A ∩ B) = 0.47 + 0.57 - 0.2679 = 0.7721

Thus, the probabilities for Dave and Mike showing up are:

P(Dave and Mike both will show up) = 0.2679 = 26.79%
P(At least one of them will show up) = 0.7721 = 77.21%

Learn more about probability here:

brainly.com/question/1210781