A national health organization predicts that 20% of

American adults will get the flu this season. If a
sample of 100 adults is selected from the population,
a. What is the probability that at least 25 of the people
will be diagnosed with the flu? (Be careful: “at
least 25” means “25 or more.”)
b. What is the probability that fewer than 15 of the
people will be diagnosed with the flu? (Be careful:
“fewer than 15” means “14 or less.”)

To solve these probability questions, we will use the binomial probability formula. Let's break down the steps to find the answers:

a. What is the probability that at least 25 of the people will be diagnosed with the flu?

Step 1: Identify the variables.
- The probability of having the flu in the population (p) = 0.20 (20%)
- The sample size (n) = 100
- The minimum number of people diagnosed with the flu (r) = 25 or more

Step 2: Calculate the probability using the binomial probability formula.
The formula for the binomial probability is given by:

P(x) = (nCx) * (p^x) * (q^(n-x))

Where:
- P(x): Probability of x successes
- nCx: The number of combinations of n items taken x at a time
- p: Probability of a success (having the flu)
- q: Probability of a failure (not having the flu), calculated as q = 1 - p
- x: The exact number of successes (people diagnosed with the flu)
- n: Sample size

To find the probability of "at least 25," we need to calculate the probability of 25 people, 26 people, 27 people, and so on, up to 100.

P(at least 25) = P(25) + P(26) + P(27) + ... + P(100) --> Calculate the sum of these probabilities.

Alternatively, we can find the complement of "at least 25" people diagnosed with the flu, which is "less than 25" people diagnosed. Then, we can subtract the probability of "less than 25" from 1 to get the probability of "at least 25."

P(at least 25) = 1 - P(less than 25) --> Find the probability of "less than 25" people diagnosed with the flu.

Now, let's calculate step by step.

Step 3: Calculate the probability of having the flu (p) and not having the flu (q).
p = 0.20 (given)
q = 1 - p = 1 - 0.20 = 0.80

Step 4: Calculate the probability of "less than 25" people diagnosed with the flu.
P(less than 25) = P(0) + P(1) + P(2) + ... + P(24) --> Sum up the probabilities from 0 to 24.

Step 5: Calculate each individual term using the binomial probability formula.
P(x) = (100Cx) * (0.20^x) * (0.80^(100-x))

For example, P(0) = (100C0) * (0.20^0) * (0.80^100)

Step 6: Find the probability of "at least 25" people diagnosed with the flu.
P(at least 25) = 1 - P(less than 25)

Repeat the steps for question b.

b. What is the probability that fewer than 15 of the people will be diagnosed with the flu?

Step 1: Identify the variables.
- The probability of having the flu in the population (p) = 0.20 (given)
- The sample size (n) = 100
- The maximum number of people diagnosed with the flu (r) = 14 or less

Step 2: Calculate the probability using the binomial probability formula.
P(fewer than 15) = P(0) + P(1) + P(2) + ... + P(14) --> Sum up the probabilities from 0 to 14.

Step 3: Calculate each individual term using the binomial probability formula.
P(x) = (100Cx) * (0.20^x) * (0.80^(100-x))

For example, P(0) = (100C0) * (0.20^0) * (0.80^100)

Step 4: Find the probability of "fewer than 15" people diagnosed with the flu.

I hope this explanation helps you understand how to approach these probability questions!