Question

At a certain University it is required by state law that 3 out of 4 students be from the state. If 10 students are selected at random from this university, what is the probability that 4 or more will be from outside the state?

I need to know how to simulate this as well.

Answer 1

Answer on Question #38487 - Physics, Math - Statistics and Probability

Probability that randomly selected student is from the state equals to 0.75.

Let $k$ be a random variable that equals to number of students among 10 that are from the state.

$k$ has binomial distribution with parameters $p = 0.75$, $n = 10$. Then probability that 4 or more are from outside the state equals to

$$\begin{array}{l} P(10 - k \geq 4) = P(k \leq 6) = 1 - P(k \geq 7) = 1 - \left(P(k = 7) + P(k = 8) + P(k = 9) + P(k = 10)\right) \\ = 1 - \left(0.75^{10} + 10 \cdot 0.25 \cdot 0.75^9 + \frac{10 \cdot 9}{2} \cdot 0.25^2 \cdot 0.75^8 + \frac{10 \cdot 9 \cdot 8}{6} \cdot 0.25^3 \cdot 0.75^7\right) \\ = 0.224 \end{array}$$

To get this result using simulation you should do following:

1) Generate 10 random numbers from 0 to 1

2) Calculate number of numbers that are less than 0.75

3) Repeat steps 1-2 T times (T is a big number, 100000 for example) and calculate number of times where results in (2) is no more than 6.

4) Divide results of (3) by T.