burgerandcheese
Junior Member
- Joined
- Jul 2, 2018
- Messages
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Question: In a survey carried out in a certain school, it is found that 3 out of 7 students passed the Mathematics test. If 8 students from that school are chosen at random, calculate the probability that
i. Exactly 5 students passed the test
ii. At least 3 students passed the test
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I just read back on binomial distribution and it said
"Bernoulli trials have the following three properties:
1. Each trial has only two possible outcomes, success and failure.
2. On every trial, each outcome has a fixed probability of occurring. If a success has a probability of p, then a failure has a probability of 1 - p.
3. The trials are independent of each other. The outcome of one trial has no influence over the outcome of another trial."
So if we do treat this question as a binomial probability, then the probability of each trial must be the same and therefore it's possible to pick the same student multiple times. In other words, the students are chosen with replacement. Does "8 students from that school are chosen at random" imply so?
This is what I did for part (i) using the the Binomial Distribution formula:
P(X = r) = C(n, r) * (p^r) * [q^(n - r)], where r = 0, 1, 2, 3, ..., n and p + q = 1
number of trials, n = 8
probability of success in each trial, p = 3/7
probability of failure in each trial, q = 1 - p = 4/7
number of successful trials, r = 5
P(X = 5) = C(8, 5) * (3/7)⁵ * (4/7)³
..= 0.1511 (4 significant figures)
i. Exactly 5 students passed the test
ii. At least 3 students passed the test
--------------------------------------------
I just read back on binomial distribution and it said
"Bernoulli trials have the following three properties:
1. Each trial has only two possible outcomes, success and failure.
2. On every trial, each outcome has a fixed probability of occurring. If a success has a probability of p, then a failure has a probability of 1 - p.
3. The trials are independent of each other. The outcome of one trial has no influence over the outcome of another trial."
So if we do treat this question as a binomial probability, then the probability of each trial must be the same and therefore it's possible to pick the same student multiple times. In other words, the students are chosen with replacement. Does "8 students from that school are chosen at random" imply so?
This is what I did for part (i) using the the Binomial Distribution formula:
P(X = r) = C(n, r) * (p^r) * [q^(n - r)], where r = 0, 1, 2, 3, ..., n and p + q = 1
number of trials, n = 8
probability of success in each trial, p = 3/7
probability of failure in each trial, q = 1 - p = 4/7
number of successful trials, r = 5
P(X = 5) = C(8, 5) * (3/7)⁵ * (4/7)³
..= 0.1511 (4 significant figures)