On any given day, a certain machine has either no malfunctions, or exactly one m

Question

On any given day, a certain machine has either no malfunctions, or exactly one malfunction. The probability of a malfunction on any given day is 0.45. Machine malfunctions on different days are independent events. 3) a. Calculate the probability that the machine has its third malfunction on the fifth day b. Calculate the probability that the machine has not had three malfunctions in the first three days. Calculate the probability that the machine has its third malfunction on the fifth day, given that the machine has not had three malfunctions in the first three days. c.

Explanation / Answer

3)

P(M) =0.45 , P(NM) = 0.55

a.

P(2 malfunction in 4 days) = C(4,2)* (0.45^2)*(0.55^2)

= 6*0.06125 = 0.3675

P(3rd malfunction in 5th day) = 0.3675*0.45 =0.1653

b.

P(3 malfunction in 3 days ) = 0.45^3 = 0.0911

P(no 3 malfunction in 3 days ) = 1- P(3 malfunction in 3 days ) = 0.909

c.

It has two part to it

i. Machine has had 1 malfunction in first 3 days

C(3,1)*0.45*0.55^2 = 3*0.136 =0.408

2nd malfunction on day 4 and 3rd on day 5 = 0.408*0.45*0.45 = 0.082

ii. Machine has had 2 malfunction in first 3 days

C(3,2)*0.45^2*0.55 = 3*0.111 =0.333

No mal function on day 4 and 3rd on day 5 = 0.333*0.55*0.45 = 0.082

P(3rd malfunction in day 5|Not 3 malfunction in first 3 days ) = i+ii = 0.082+0.082 = 0.164

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