R. H. Bruskin Associates Market Research found that 30% of Americans do not think that having a

college education is important to succeed in the business world. If a random sample of five Americans
is s selected, find these probabilities.
(a) Exactly two people will agree with that statement.

(b) At most three people will agree with that statement.

(c) At least two people will agree with that statement.

(d) Fewer than three people agree with that statement

If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.

a) .3^2 * .7^3 = ?

b) Same as 1, 2 or 3 people agreeing.

for one: .3 * .7^4 = ?
for two: see above
for three: .3^3 * .7^2 = ?

Either-or probabilities are found by adding the individual probabilities.

c) Same as 2, 3, 4, or 5 agreeing.

d) Same as 1 or 2 agreeing.

To find the probabilities, we will use the binomial probability formula:

P(x) = C(n, x) * p^x * (1-p)^(n-x)

where:
P(x) is the probability of getting exactly x successes
n is the number of trials
x is the number of successes
p is the probability of success for a single trial
1-p is the probability of failure for a single trial

In our case:
n = 5 (number of Americans in the sample)
p = 0.3 (probability of an American agreeing with the statement)
1-p = 0.7 (probability of an American disagreeing with the statement)

(a) To find the probability that exactly two people will agree with the statement, we substitute x = 2 into the formula:

P(2) = C(5, 2) * 0.3^2 * 0.7^3

C(5, 2) = 5! / (2! * (5-2)!) = 5! / (2! * 3!) = 10

P(2) = 10 * 0.3^2 * 0.7^3 = 0.3087 (rounded to 4 decimal places)

Therefore, the probability that exactly two people will agree with the statement is approximately 0.3087.

(b) To find the probability that at most three people will agree with the statement, we need to calculate the probabilities for x = 0, 1, 2, and 3, and then sum them up.

P(at most 3) = P(0) + P(1) + P(2) + P(3)

P(0) = C(5, 0) * 0.3^0 * 0.7^5 = 1 * 1 * 0.7^5 = 0.16807
P(1) = C(5, 1) * 0.3^1 * 0.7^4 = 5 * 0.3 * 0.7^4 = 0.36015
P(2) = 0.3087 (from part a)
P(3) = C(5, 3) * 0.3^3 * 0.7^2 = 10 * 0.3^3 * 0.7^2 = 0.1323

P(at most 3) = 0.16807 + 0.36015 + 0.3087 + 0.1323 = 0.96922 (rounded to 5 decimal places)

Therefore, the probability that at most three people will agree with the statement is approximately 0.96922.

(c) To find the probability that at least two people will agree with the statement, we need to calculate the probabilities for x = 2, 3, 4, and 5, and then sum them up.

P(at least 2) = P(2) + P(3) + P(4) + P(5)

P(4) = C(5, 4) * 0.3^4 * 0.7^1 = 5 * 0.3^4 * 0.7^1 = 0.07203
P(5) = C(5, 5) * 0.3^5 * 0.7^0 = 1 * 0.3^5 * 0.7^0 = 0.00243

P(at least 2) = 0.3087 + 0.1323 + 0.07203 + 0.00243 = 0.51546 (rounded to 5 decimal places)

Therefore, the probability that at least two people will agree with the statement is approximately 0.51546.

(d) To find the probability that fewer than three people agree with the statement, we need to calculate the probabilities for x = 0, 1, and 2, and then sum them up.

P(fewer than 3) = P(0) + P(1) + P(2)

P(fewer than 3) = 0.16807 + 0.36015 + 0.3087 = 0.83692 (rounded to 5 decimal places)

Therefore, the probability that fewer than three people agree with the statement is approximately 0.83692.

To find the probabilities for each scenario, we can use the binomial probability formula:

P(x) = nCx * p^x * (1 - p)^(n - x)

Where:
- P(x) is the probability of getting exactly x successes
- nCx is the combination formula for selecting x items from n items
- p is the probability of success for each trial
- (1 - p) is the probability of failure for each trial
- n is the total number of trials

In this case, we have a binomial distribution where the probability of success (p) is 0.3 (the percentage of Americans who do not think college education is important) and the total number of trials (n) is 5 (the number of randomly selected Americans). Let's calculate the probabilities for each scenario:

(a) Exactly two people will agree with that statement.
To find this probability, we substitute x = 2 into the formula:
P(2) = 5C2 * (0.3^2) * (0.7^3)

(b) At most three people will agree with that statement.
To find this probability, we need to calculate the probabilities for x = 0, 1, 2, and 3, and then sum them up:
P(0) + P(1) + P(2) + P(3) = 5C0 * (0.3^0) * (0.7^5) + 5C1 * (0.3^1) * (0.7^4) + P(2) + P(3)

(c) At least two people will agree with that statement.
To find this probability, we need to calculate the probabilities for x = 2, 3, 4, and 5, and then sum them up:
P(2) + P(3) + P(4) + P(5)

(d) Fewer than three people agree with that statement.
To find this probability, we need to calculate the probabilities for x = 0, 1, and 2, and then sum them up:
P(0) + P(1) + P(2)

Now let's substitute the values and perform the calculations to find the actual probabilities.