Answer to Question #229507 in Statistics and Probability for Fatma Al Hashimi

Question

The probability that a student is admitted to MCBS for bachelor is 0.6. If 10 students from the same school apply, find the probability that (i) at most 5 are admitted. (ii) at least 7 are rejected

Answer 1

Let $X=$ the number of admitted students: $X\sim Bin(n, p).$

Given $p=0.6, n=10, q=1-p=0.4.$

(i)

$$P(X\leq5)=P(X=0)+P(X=1)+P(X=2)$$

$$+P(X=3)+P(X=4)+P(X=5)$$

$$=\dbinom{10}{0}(0.6)^0(0.4)^{10-0}+\dbinom{10}{1}(0.6)^1(0.4)^{10-1}$$

$$+\dbinom{10}{2}(0.6)^2(0.4)^{10-2}+\dbinom{10}{3}(0.6)^3(0.4)^{10-3}$$

$$+\dbinom{10}{4}(0.6)^4(0.4)^{10-4}+\dbinom{10}{5}(0.6)^5(0.4)^{10-5}$$

$$\approx0.3667$$

(ii)

$$P(X\leq3)=P(X=0)+P(X=1)+P(X=2)$$

$$+P(X=3)=\dbinom{10}{0}(0.6)^0(0.4)^{10-0}$$

$$+\dbinom{10}{1}(0.6)^1(0.4)^{10-1}+\dbinom{10}{2}(0.6)^2(0.4)^{10-2}$$

$$+\dbinom{10}{3}(0.6)^3(0.4)^{10-3}\approx0.0548$$