In a certain college, 33% of the physics majors belong to ethnic minorities. if 10 students are selected at random from the physics majors, what is the probability that no more than 6 belong to an ethnic minority? round to three decimal places.

In a certain college, 33% of the physics majors belong to ethnic minorities. 10 students are selected at random from physics majors.

Let x be the number of students selected from physics majors belongs to ethnic minorities.

Let p be the probability that selected student belongs to ethnic minorities. Let n be the total number of students selected from physics majors.

Given: n=10 and p=0.33

X follows Binomial distribution with n=10 and p=0.33

The probability that no more than 6 belongs to ethnic majority

P(x ≤ 6) = P(x=6) + p(x=5)+P(x=4) + p(x=3) + P(x=2) + p(x=1) + p(x=0)

Using Binomial distribution to find the probabilities

P(x=k) =

P(x=6) =

P(x=6) = 0.0546

P(x=5) =

P(x=5) = 0.1332

P(x=4) =

P(x=4) = 0.2253

P(x=3) =

P(x=3) = 0.2613

P(x=2) =

P(x=2) = 0.1989

P(x=1) =

P(x=1) = 0.0897

P(x=0) =

P(x=0) = 0.0182

P(x ≤ 6) = 0.0546 + 0.1332+ 0.2253 +0.2613+ 0.1989 + 0.0897 + 0.0182

P(x ≤ 6) = 0.9821 ~ 0.982

The probability that no more than 6 belong to an ethnic minority is 0.982

Using the binomial distribution, it is found that there is a 0.982 = 98.2% probability that no more than 6 belong to an ethnic minority

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For each student, there are only two possible outcomes. Either they belong to  an ethnic minority, or they do not. The probability of a student belonging to an ethnic minority is independent of any other student, which means that the binomial probability distribution is used to solve this question.

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Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

In which is the number of different combinations of x objects from a set of n elements, given by the following formula.

And p is the probability of a success on a single trial.

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In which

Thus

Then

0.982 = 98.2% probability that no more than 6 belong to an ethnic minority

A similar problem is given at link

the answer is 24

step-by-step explanation: