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 breakdown.

Answer 1

Let $X$ be number of breakdowns, $p=0.2, \ n=5.$ Then it is the random variable with the binomial distribution: $X \backsim B(n,p).$

The probability of having $k$ breakdowns is $P(X=k)=\binom{n}{k} p^k(1-p)^{n-k}$

(I) Thee are no breakdowns: $P(X=0)=\binom{5}{0}\times0.2^0\times 0.8^5=0.8^5=0.32768$

(II) Four machines break down: $P(X=4)=\binom{5}{4}\times 0.2^4\times 0.8^1=5\times 0.2^4\times 0.8=0.0064$

(III) All machines break down: $P(X=5)=\binom{5}{5} \times 0.2^5\times 0.8^0=0.2^5=0.00032$

(IV) At least three machines breakdown: $P(X\geq3)=P(X=3)+P( X=4)+P(X=5)=\binom {5}{3}\times 0.2^3\times 0.8^2+0.0064+0.00032=10\times 0.2^3\times 0.8^2+0.00672=0.05792$

Answer: (I) 0.32768, (II) 0.0064, (III) 0.00032, (IV) 0.05792.