Suppose that 90% of all batteries from a certain supplier have accept able volta

Question

Suppose that 90% of all batteries from a certain supplier have accept able voltages and that batteries are independent. A certain type of flashlight requires two type-D batteries, and the flashlight will work only if both its batteries have acceptable voltages. (a) Calculate the probability that a randomly selected flashlight works (you may assume there are batteries in the flashlight). (b) Among twenty-seven (27) randomly selected flashlights, what is the probability that (i) exactly twenty-two (22) andi) at least twenty (20) will work?

Explanation / Answer

a. It is said that a flashlight will work, only if both its batteries have acceptable voltages. It is also known that 90% of all batteries from a certain supplier have acceptable volatges and batteries are independent. Thus, using multiplication rule [if A and B are independent events, then the probability of A and B is P(A and B)=P(A)*P(B)], the probability that a randomly selected flashlight works is 0.90*0.90=0.81

b. There are n=27 independent trials, with probability of success, p=0.90. The probability of success is constant throughout the trials. This accounts for binomial distribution for Bernoulli trials. Use, P(X,r)=nCr(p)^r(1-p)^n-r, where, r denotes specific number of success in n trials.

i) P(X=22)=27C22(0.90)^22(1-0.90)^5=0.0795

ii) P(X>=20)=1-P(X<20)=1-P(X<=19)=1-0.002999 [use binomial distribution table, for n=27, p=0.9, x=19, the table gives cumulative probability, P(X<=x)]

=0.997001

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