In a Gallop Poll, 35% 18-23 year old's stated that they believe in ghosts. Find the probability that out of 16 students aged 18-23:

A. Exactly 5 said they believed in ghosts.
B. Exactly 5 said they do not believe in ghosts.
C.Less than 11 said they do not believe in ghosts.
D. At least 8 said they believe in ghosts.

p=.35 q =.65

.35^5 times .65^11 =_____

then you have to multiply that answer by the combination that gets you 16!/11!5! you do that because there are many ways to get those 5 people. It could be the first 5, the last 5, the middle 5 or any possible.

b) .35^11 times .65^5 and then multiply by the same combination that you had above.

c) less than 11 means 0-10 or you can look at it as 11-16 and you would subtract that answer from 1.
This is very complicated and takes a long time to do each possibility. Are you allowed to use a TI-84 calculator or something similar?

At least 8 means 8 -16
.35^8 times .65^8 + .35^9 times .65^7 +.....+ .35^16 times .65^0

for each term you will need the combination that is 16! divided by the factorial of the exponents in each case. Very tedious... so a calculator will help.

To solve these probability questions, we'll use the binomial probability formula:

P(X=k) = nCk * p^k * q^(n-k)

where n is the total number of trials, k is the number of successful trials, p is the probability of success, q is the probability of failure (1-p), and nCk represents the number of ways to choose k successes from n trials.

Given:
p = 0.35 (probability of believing in ghosts)
q = 1 - p = 0.65 (probability of not believing in ghosts)
n = 16 (total number of students aged 18-23)

A. Exactly 5 said they believed in ghosts.
Using the binomial probability formula:
P(X=5) = 16C5 * (0.35)^5 * (0.65)^(16-5)

Calculating this, we get:
P(X=5) ≈ 0.2633 or 26.33%

B. Exactly 5 said they do not believe in ghosts.
Since the probability of not believing in ghosts is the complement of believing in ghosts, we use the same formula as in part A:
P(X=5) = 16C5 * (0.65)^5 * (0.35)^(16-5)

Calculating this, we get:
P(X=5) ≈ 0.2538 or 25.38%

C. Less than 11 said they do not believe in ghosts.
To find the probability that less than 11 said they do not believe in ghosts, we sum up the probability of each case where k ranges from 0 to 10 using the binomial probability formula:

P(X<11) = Σ(k=0 to 10) (16Ck * (0.65)^k * (0.35)^(16-k))

Calculating this, we get:
P(X<11) ≈ 0.9573 or 95.73%

D. At least 8 said they believe in ghosts.
To find the probability that at least 8 said they believe in ghosts, we add up the probabilities of cases where k ranges from 8 to 16 using the binomial probability formula:

P(X≥8) = Σ(k=8 to 16) (16Ck * (0.35)^k * (0.65)^(16-k))

Calculating this, we get:
P(X≥8) ≈ 0.9974 or 99.74%

To solve this problem, we need to use the binomial distribution formula. The binomial distribution is a discrete probability distribution that gives the probability of a certain number of successes in a fixed number of independent Bernoulli trials (experiments with only two possible outcomes: success or failure).

In this case:
- The probability of success (p) is 35% or 0.35 (those who believe in ghosts).
- The probability of failure (q) is 65% or 0.65 (those who do not believe in ghosts).
- The total number of trials (n) is 16 students.

A. To find the probability that exactly 5 students said they believed in ghosts, we can use the binomial distribution formula:

P(X = k) = C(n, k) * p^k * q^(n-k)

where:
- P(X = k) is the probability of exactly k successes
- C(n, k) is the number of possible combinations of selecting k successes from n trials, which can be calculated as C(n, k) = n! / (k!(n-k)!)

So, for this case:
P(X = 5) = C(16, 5) * (0.35)^5 * (0.65)^(16-5)

B. To find the probability that exactly 5 students said they do not believe in ghosts, we can use the same formula as in part A but with p and q swapped:

P(X = k) = C(n, k) * q^k * p^(n-k)

So, for this case:
P(X = 5) = C(16, 5) * (0.65)^5 * (0.35)^(16-5)

C. To find the probability that less than 11 students said they do not believe in ghosts, we can sum up the probabilities for k ranging from 0 to 10 using the binomial distribution formula from part B:

P(X < 11) = P(X = 0) + P(X = 1) + ... + P(X = 10)

D. To find the probability that at least 8 students said they believe in ghosts, we need to calculate the complementary probability of at most 7 students believing in ghosts. We can use the binomial distribution formula from part A to calculate the probability for k ranging from 0 to 7 and subtract it from 1:

P(X ≥ 8) = 1 - (P(X = 0) + P(X = 1) + ... + P(X = 7))

Using these formulas, you can calculate the probabilities for each scenario based on the given information.

it should be D.