Answer to Question #107761 in Statistics and Probability for Emmanuel Daniels

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

Let p be the probability of a ATM breakdown,

p=0.2

i. there are no breakdowns,

$P_1=(1-p)^5\\ P_1=(0.8)^5\\ P_1=0.32768$

ii. four machines break down

$P_2=(p^4 )\times (1- p)\\ P_2=(0.2)^4 \times (0.8)\\ P_2=0.00128$

iii. all machines break down

$P_3=p^5\\ P_3=(0.2)^5\\ P_3=0.00032$

iv. at least three machines break down

$P_4=P_2+P_3+(p^3\times(1-p)^2)\\ P_4=0.00128+0.00032+0.2^3\times0.8^2\\ P_4=0.00672$