1) If the probability that a tennis player will serve an ace is .27, what is the probability that he will serve: (only set up. DO NOT evaluate.)
a) exactly 3 aces out of 8 serves?
b) at least 1 ace out of 8 serves?
a) p(x=3) (8,3) (.27)^3 (.73)^5
b) At least ---> no more p (x > or equal to 1)
= 1 - sum of p(x+1) and p(x+3) (please correct me if I'm wrong)
2) A survey shows that 75% of the households in a large town have microwave ovens. If 20 houses are selected at random, find the following (only set up. DO NOT evalute)
a) The probability that five of these households have microwave ovens.
b) The probability that no more than one of these households has a microwave oven.
c) The probability that at least 3 of these households have a microwave oven.
a) p(x=5) = x(20,5) (.75)^5 (.25)^15
b)no more---> p(x< or equal to 1)
p(x< or equal to 1) = p(x+0) + p(x+1)
c) P(x> or equal to 3)
= 1 - p(x=0)+p(x=1)+p(x=2)
To calculate the probability of certain events in these scenarios, we can use the binomial probability formula:
P(x) = (n C x) * p^x * (1-p)^(n-x)
Where:
- P(x) represents the probability of x events occurring
- n represents the total number of trials/serves
- x represents the number of successful/ace serves
- p represents the probability of a successful/ace serve
- (n C x) is the combination formula, which calculates the number of ways we can select x successes out of n trials
Now let's set up the probabilities for each question:
Question 1:
a) To find the probability of getting exactly 3 aces out of 8 serves, we can use the formula:
P(x=3) = (8 C 3) * (0.27)^3 * (0.73)^5
b) To find the probability of getting at least 1 ace out of 8 serves, we need to consider the complement probability. The complement of getting at least 1 ace is getting no aces, so we can calculate the probability of getting no aces and subtract it from 1:
P(no ace) = P(x=0) = (8 C 0) * (0.27)^0 * (0.73)^8
P(at least 1 ace) = 1 - P(no ace)
Question 2:
a) To find the probability that five of the 20 households have microwave ovens, we can use the formula:
P(x=5) = (20 C 5) * (0.75)^5 * (0.25)^15
b) To find the probability that no more than one household has a microwave oven, we can find the probability of getting 0 or 1 households with ovens:
P(x< or equal to 1) = P(x=0) + P(x=1)
c) To find the probability that at least 3 households have a microwave oven, we can find the complement probability of having 0, 1, or 2 households with ovens:
P(no more than 2 households with ovens) = P(x=0) + P(x=1) + P(x=2)
P(at least 3 households with ovens) = 1 - P(no more than 2 households with ovens)
Remember, these are only the setup of the probabilities. To get the actual values, you need to evaluate the binomial formulas using the correct values for n, x, and p.