Question

They sell lottery tickets at the kiosk. A single lottery ticket wins with a probability of 0.45. With

likelihood

a) all bets win 

b) at least one lot wins 

c) exactly two lotteries win?

Answer 1

Let $X=$ the number of tickets which win: $X\sim Bin (n, p)$

Given $p=0.45$

a)

$$P(X=n)=\dbinom{n}{n}(0.45)^n(1-0.45)^{n-n}=(0.45)^n$$

b)

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

$$=1-\dbinom{n}{0}(0.45)^0(1-0.45)^{n-0}=1-(0.55)^n$$

c)

$$P(X=2)=\dbinom{n}{2}(0.45)^2(1-0.45)^{n-2}$$

$$=\dfrac{n(n-1)}{2}(0.45)^2(0.55)^{n-2}$$