Mathematics, 14.07.2019 21:30 xcncxgnfxg6487

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.

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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) = $(10Ck) 0.33^{k} (1-0.33)^{10-k}$

P(x=6) = $(10C6) 0.33^{6} (1-0.33)^{10-6}$

P(x=6) = 0.0546

P(x=5) = $(10C5) 0.33^{5} (1-0.33)^{10-5}$

P(x=5) = 0.1332

P(x=4) = $(10C4) 0.33^{4} (1-0.33)^{10-4}$

P(x=4) = 0.2253

P(x=3) = $(10C3) 0.33^{3} (1-0.33)^{10-3}$

P(x=3) = 0.2613

P(x=2) = $(10C2) 0.33^{2} (1-0.33)^{10-2}$

P(x=2) = 0.1989

P(x=1) = $(10C1) 0.33^{1} (1-0.33)^{10-1}$

P(x=1) = 0.0897

P(x=0) = $(10C0) 0.33^{0} (1-0.33)^{10-0}$

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

Answer from: Javanese5987

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.

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

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

$C_{n,x} = \frac{n!}{x!(n-x)!}$

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

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33% are ethnic minorities, thus, p = 0.33

10 students are chosen, thus, n = 10

.The probability that no more than 6 are ethnic minorities is:

$P(X \leq 6) = 1 - P(X 6)$

In which

P(X 6) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)

Thus

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 7) = C_{10,7}.(0.33)^{7}.(0.67)^{3} = 0.0154$

$P(X = 8) = C_{10,8}.(0.33)^{8}.(0.67)^{2} = 0.0028$

$P(X = 9) = C_{10,9}.(0.33)^{9}.(0.67)^{1} = 0.0003$

$P(X = 10) = C_{10,9}.(0.33)^{10}.(0.67)^{0} \approx 0$

Then

P(X 6) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) = 0.0154 + 0.0028 + 0.0003 + 0 = 0.0185

$P(X \leq 6) = 1 - P(X 6) = 1 - 0.0185 = 0.9815$

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

A similar problem is given at link

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on a coordinate grid, point a is at (−4.0, −2.4) and point b is at (4.0, −2.4). point b is a reflection of point a across the –axis. input either a lowercase x or i think is the right awnser

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0.5?

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In a certain college, 33% of the physics majors belong to ethnic minorities. if 10 students are sele...