suppose the probability is 0.30 that a student taking an exam did not review. what is the probability,that among 6 students taking an exam, 5 did not review? less than or equal to 4 did not review?
To find the probability that a certain number of students did not review, we can use the binomial probability formula.
The binomial probability formula is:
P(X=k) = (n C k) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of k successes,
- n is the total number of trials,
- k is the number of successful outcomes,
- p is the probability of success in a single trial,
- (n C k) is the binomial coefficient, which represents the number of ways to choose k successes from n trials.
Let's calculate the probabilities for the given cases:
1. Probability that exactly 5 students did not review:
- n = 6 (total number of students)
- k = 5 (number of students who did not review)
- p = 0.30 (probability that a student did not review)
Calculating the binomial probability:
P(X=5) = (6 C 5) * 0.30^5 * (1-0.30)^(6-5)
(6 C 5) = 6! / (5! * (6-5)!)
= 6
P(X=5) = 6 * 0.30^5 * (1-0.30)^(6-5)
= 6 * 0.30^5 * 0.70^1
= 6 * 0.00243 * 0.70
≈ 0.010206
So, the probability that exactly 5 students did not review is approximately 0.010206.
2. Probability that less than or equal to 4 students did not review:
To find this probability, we need to calculate the sum of probabilities for k = 0, 1, 2, 3, and 4.
- n = 6 (total number of students)
- p = 0.30 (probability that a student did not review)
Calculating the individual probabilities:
P(X=0) = (6 C 0) * 0.30^0 * (1-0.30)^(6-0)
= 1 * 1 * 0.70^6
≈ 0.117649
P(X=1) = (6 C 1) * 0.30^1 * (1-0.30)^(6-1)
= 6 * 0.30 * 0.70^5
≈ 0.302526
P(X=2) = (6 C 2) * 0.30^2 * (1-0.30)^(6-2)
= 15 * 0.30^2 * 0.70^4
≈ 0.324135
P(X=3) = (6 C 3) * 0.30^3 * (1-0.30)^(6-3)
= 20 * 0.30^3 * 0.70^3
≈ 0.18522
P(X=4) = (6 C 4) * 0.30^4 * (1-0.30)^(6-4)
= 15 * 0.30^4 * 0.70^2
≈ 0.059535
Summing up the probabilities:
P(X≤4) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)
≈ 0.117649 + 0.302526 + 0.324135 + 0.18522 + 0.059535
≈ 0.989065
So, the probability that less than or equal to 4 students did not review is approximately 0.989065.