Question

There are five old ATM machines at Smart Bank. All probability that an ATM breaks down in the course of a day is 0.2. If they are all in working order at the beginning a day, and breakdowns are independent of each other, find the probability that during the day; i. there are no breakdowns ii. four machines break down iii. all machines break down iv. at least three machines break down

Answer 1

i. This is a binomial distribution with p=0.2, n=5.

$P(X=0)=C^0_5p^0(1-p)^5=(1-0.2)^5=0.32768.$

ii. $P(X=4)=C^4_5p^4(1-p)^1=5*0.2^4*(1-0.2)^1=0.0064.$

iii. $P(X=5)=C^5_5p^5(1-p)^0=0.2^5=0.00032.$

iv. $P(X\ge3)=P(X=3)+P(X=4)+P(X=5)=$

$=C^3_5p^3(1-p)^2+C_5^4p^4(1-p)^1+C_5^5p^5(1-p)^0=$

$=10*0.2^3*(1-0.2)^2+5*0.2^4+(1-0.2)^1+0.2^5=0.05792.$