An insurance agent insures lives of 5 men, all of the same age and in good health. The probability that a man of this age will survive tha next 30 years is known to be `(2)/(3)`. The probability that in the next 30 years at most three men will survive is

Explanation

Let x = number of men who survive the next 30 years
P = probability that the man will suvive the next 30 years `therefore p = 2/3 `
` therefore q= 1 - p =1 - 2/3 = 1/3 `
Given : n=5
` therefore X~ B (5, 2/3 )`
The p.m on X is given as : `P [X= x] = ""^nC_xp^xq^(n-x) `
i.e `p(x)= ""^5C_x(2/3)^x(1/3)^(5-x)`
P (at most three men will suvive in the next 30 years )
`= P [X le 3 ] 1-P [X gt 3]`
`=1-[P (X =4)+P( X = 5)`
`= 1- p(4) + p(5)]`
`1-[""^5C_4(2/3)^4(1/2)^(5-4)+""^5C_5(2/3)^5(1/3)^(5-5)]`
`1-[5xx 16/81xx1/3+1xx32/243 ]`
`=1-[(80+32)/243]=1-112/243`
`=1- 0.4609 = 0.5391 `
Hence , the probability that in the next 30 years at most 3 men will survive is 0.5391.