The probability that a student is not swimmer is 1/5.find the probability that out of 5 students (1) at least 4 are swimmers and (2)at most 3 are swimmers

prob(swim)=4/5

prob(not swim) = 1/5
prob(at least 4 swim)
= prob(4of5 swim)+prob(5of5 swim)
= C(5,4) (4/5)^4 (1/5) + (4/5)^5
= ....

#2
how does "at most 3 are swimmers" relate to what we just found ?

Multiply on both sides mat 1/5

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Multiply on both sides mat 1/5

P (x)=1/5

At most 3 are swimmer i didn't got these

To find the probabilities, we need to first determine the probability that a student is a swimmer.

If the probability that a student is not a swimmer is 1/5, then the probability that a student is a swimmer is the complement of this, which is 1 - 1/5 = 4/5.

Now, we can proceed to find the probabilities for the given scenarios:

(1) At least 4 are swimmers:
To find the probability that at least 4 out of 5 students are swimmers, we need to consider two cases: exactly 4 swimmers and exactly 5 swimmers.

The probability that exactly 4 students are swimmers can be calculated using the binomial probability formula:

P(4 swimmers) = C(5, 4) * (4/5)^4 * (1/5)^1,
where C(5, 4) is the number of combinations of choosing 4 swimmers out of 5.

The probability that exactly 5 students are swimmers is simply (4/5)^5 since all of the students must be swimmers.

Finally, we sum up these probabilities to get the total probability of at least 4 students being swimmers.

P(at least 4 swimmers) = P(4 swimmers) + P(5 swimmers)

(2) At most 3 are swimmers:
To find the probability that at most 3 out of 5 students are swimmers, we need to consider three cases: exactly 0, 1, 2, or 3 swimmers.

The probability that exactly 0 students are swimmers is calculated by (1/5)^5 since none of the students can be swimmers.

Similarly, the probabilities for exactly 1, 2, and 3 students being swimmers can be calculated using the binomial probability formula.

Finally, we sum up these probabilities to get the total probability of at most 3 students being swimmers.

P(at most 3 swimmers) = P(0 swimmers) + P(1 swimmer) + P(2 swimmers) + P(3 swimmers)

By evaluating the respective formulas, you can find the numerical values for the probabilities in both scenarios.