Question

. There are five 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, find 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

Let $X=$ the number of machines which are break down: $X\sim Bin (n;p)$

$$P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$$

Given that $n=5, p=0.2$

i. there are no breakdowns 

$$P(X=0)=\binom{5}{0}(0.2)^0(1-0.2)^{5-0}=$$ $$=(0.8)^5=0.32768$$

ii. four machines break down 

$$P(X=4)=\binom{5}{4}(0.2)^4(1-0.2)^{5-4}=$$ $$=5(0.2)^4 (0.8)=0.0064$$

iii. all machines break down 

$$P(X=5)=\binom{5}{5}(0.2)^5(1-0.2)^{5-5}=$$ $$=(0.2)^5=0.00032$$

iv. at least three machines break down

$$P(X\geq 3)=P(X=3)+P(X=4)+P(X=5)=$$ $$=\binom{5}{3}(0.2)^3(1-0.2)^{5-3}+0.0064+0.00032=$$ $$=10(0.2)^3(0.8)^2 +0.00672=0.05792$$