Question

An athlete is running in five races and in each race he has a 70% chance of winning. What is the probability that he will win at least two races?

Answer 1

Let $X=$ the number of winned races: $X\sim Bin(n, p)$

Given $n=5, p=0.7$

$$P(X\geq2)=1-P(X=0)-P(X=1)$$

$$=1-\dbinom{5}{0}(0.7)^0(1-0.7)^{5-0}-\dbinom{5}{1}(0.7)^1(1-0.7)^{5-1}$$

$$=1-(0.3)^5-5(0.7)(0.3)^4$$

$$=1-3.8(0.3)^4=0.96922$$

The probability that he will win at least two races is 0.96922.