26, From 2022/2023 academic year Statistics, the probabiliry that a freshman entering AMU
(College of agricultural sciences) wil1 survive fist semester is 0.92. Assuming this patten
remain unchanged over the subsequent years, what is the probability that among 100
randonly selected freshmen in first semester.
a) None will survive?
b) Exactly 97 will survive?
c) At least three will survive?
To solve this problem, we can use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- n is the number of trials (number of freshmen in this case)
- k is the number of successful events (number of freshmen surviving in this case)
- p is the probability of success (probability of a freshman surviving in this case)
For each part of the problem, we will substitute the appropriate values into the formula.
a) None will survive (k = 0):
P(X = 0) = (100 choose 0) * (0.92)^0 * (1 - 0.92)^(100 - 0)
= 1 * 1 * 0.08^100
≈ 1 * 1 * 7.8886e-08
≈ 7.8886e-08
So, the probability that none of the 100 randomly selected freshmen will survive the first semester is approximately 7.8886e-08.
b) Exactly 97 will survive (k = 97):
P(X = 97) = (100 choose 97) * (0.92)^97 * (1 - 0.92)^(100 - 97)
= (100! / (97! * 3!)) * (0.92)^97 * (0.08)^3
≈ (100 * 99 * 98 / (3 * 2 * 1)) * 0.036 ^ 3 * 0.08 ^ 3
≈ 161700 * 0.00046 * 0.000512
≈ 0.0389376
So, the probability that exactly 97 out of 100 randomly selected freshmen will survive the first semester is approximately 0.0389376.
c) At least three will survive (k ≥ 3):
To find this probability, we need to calculate the probability of three or more freshmen surviving and subtract it from 1:
P(X ≥ 3) = 1 - P(X < 3)
= 1 - [P(X = 0) + P(X = 1) + P(X = 2)]
= 1 - [0.08^100 + (100 choose 1) * (0.92)^1 * (0.08)^(100-1) + (100 choose 2) * (0.92)^2 * (0.08)^(100-2)]
Using the binomial probability formula for P(X = 0), P(X = 1), and P(X = 2), we can calculate:
P(X ≥ 3) ≈ 1 - [7.8886e-08 + (100 * 0.92 * 0.08^99) + (100 * 99 / 2) * (0.92)^2 * (0.08)^98]
≈ 1 - [7.8886e-08 + 0.07345396147 + 0.2540021655]
≈ 0.67254394883
So, the probability that at least three out of 100 randomly selected freshmen will survive the first semester is approximately 0.67254394883.