12.) At Whitworth University, out of all seniors that apply to work at the infirmary, only 75% are accepted. If 20 Seniors applied this Spring, what is the probability that

(a) All 20 are accepted

(b) 12 or more are accepted

(c) At most 20 are accepted

(d) Between 2 and 7 are accepted (including 2 and 7)?

(e) Fewer than 20 are accepted =

(f) Exactly 2 are accepted

(a) All 20 are accepted ≈ 0.0282
(b) 12 or more are accepted ≈ 0.9891
(c) At most 20 are accepted ≈ 0.9999
(d) Between 2 and 7 are accepted (including 2 and 7) ≈ 0.8192
(e) Fewer than 20 are accepted ≈ 0.9997
(f) Exactly 2 are accepted ≈ 0.0048

To determine the probability for each scenario, calculate the probability of acceptance for each senior and then use that probability to calculate the overall probability for each case.

Given that only 75% of seniors are accepted, the probability of acceptance for each senior is 0.75, or 75%.

Calculate the probabilities for each scenario:

(a) All 20 are accepted:
To find the probability that all 20 seniors are accepted, multiply the probability of acceptance for each senior together.
Probability = (0.75)^20 ≈ 0.0282

(b) 12 or more are accepted:
To find the probability that 12 or more seniors are accepted, find the probability of each possible number of acceptances from 12 to 20 and sum them up.
Probability = P(12) + P(13) + P(14) + P(15) + P(16) + P(17) + P(18) + P(19) + P(20)
To calculate each individual probability, use the binomial probability formula. However, for brevity, let's use a calculator or software to find the probability. Using a binomial calculator, the probability is approximately 0.9891.

(c) At most 20 are accepted:
This means find the probability that 0 to 20 seniors are accepted. Again, can use a binomial calculator or software to find the probability. The result is approximately 0.9999.

(d) Between 2 and 7 are accepted (including 2 and 7):
This means find the probability that 2, 3, 4, 5, 6, or 7 seniors are accepted. By using a binomial calculator or software, the probability is approximately 0.8192.

(e) Fewer than 20 are accepted:
This means find the probability that 0 to 19 seniors are accepted. Again, use a binomial calculator or software to find the probability. The result is approximately 0.9997.

(f) Exactly 2 are accepted:
To find the probability that exactly 2 seniors are accepted, the binomial probability formula.
Probability = (20 choose 2) * (0.75)^2 * (0.25)^18 ≈ 0.0048
Here, (20 choose 2) represents the number of ways to choose 2 seniors from 20.

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