Question

In a certain college, 33% of the physics majors belong to ethnic minorities. Find the probability of the event from a random sample of 10 students who are physics majors.

Answer 1

The probability that no more than 6 students belong to a ethnic minority is 0.9815.

Explanation:

The question is incomplete:

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?

We can model this with a random variable, with sample size n=10 and probability of success p=0.33.

The probability that k answers are guessed right in the sample is:

`P(x=k)=\dbinom{n}{k}p^k(1-p)^(n-k)=\dbinom{10}{k}\cdot0.33^k\cdot0.67^(10-k)`

We have to calculate the probability that 6 or less students belong to a ethnic minority. This can be calculated as:

`P(x\leq6)=P(x=0)+P(x=1)+P(x=2)+P(x=3)+P(x=4)+P(x=5)+P(x=6)\\\\\\`

`P(x=0)=\dbinom{10}{0}\cdot0.33^(0)\cdot0.67^(10)=1\cdot1\cdot0.0182=0.0182\\\\\\P(x=1)=\dbinom{10}{1}\cdot0.33^(1)\cdot0.67^(9)=10\cdot0.33\cdot0.0272=0.0898\\\\\\P(x=2)=\dbinom{10}{2}\cdot0.33^(2)\cdot0.67^(8)=45\cdot0.1089\cdot0.0406=0.1990\\\\\\P(x=3)=\dbinom{10}{3}\cdot0.33^(3)\cdot0.67^(7)=120\cdot0.0359\cdot0.0606=0.2614\\\\\\`

`P(x=4)=\dbinom{10}{4}\cdot0.33^(4)\cdot0.67^(6)=210\cdot0.0119\cdot0.0905=0.2253\\\\\\P(x=5)=\dbinom{10}{5}\cdot0.33^(5)\cdot0.67^(5)=252\cdot0.0039\cdot0.135=0.1332\\\\\\P(x=6)=\dbinom{10}{6}\cdot0.33^(6)\cdot0.67^(4)=210\cdot0.0013\cdot0.2015=0.0547\\\\\\\\\\\\`

`P(x\leq6)=0.0182+0.0898+0.1990+0.2614+0.2253+0.1332+0.0547\\\\\\P(x\leq6)=0.9815`

The probability that 6 or less students belong to a ethnic minority is 0.9815.