Question

Suppose a road is flooded with probability p = 0.10 during a year and not more than one flood occurs during a year.

What is the probability that it will be flooded at most twice during a 10-year period?

Answer 1

road is flooded with probability p = 0.10 during a year

not more than one flood occurs during a year

during a 10-year period, n = 10

Let, X be the number of times road is flooded during a 10-year period

probability that it will be flooded at most twice during a 10-year period

P[ X <= 2 ] = P[ X = 0 ] + P[ X = 1 ] + P[ X = 2 ]

Since, the floods are independent in occurrance.

Hence, it will follow binomial

P[ X = k ] = nCk*p^k*(1-p)^(n-k)

P[ X = 0 ] = 10C0*0.1^0*(1-0.1)^(10-0)

P[ X = 0 ] = 10C0*0.1^0*0.9^10

P[ X = 0 ] = 1*0.9^10

P[ X = 0 ] = 0.3487

P[ X = 1 ] = 10C1*0.1^1*(1-0.1)^(10-1)

P[ X = 1 ] = 10C1*0.1^1*0.9^9

P[ X = 1 ] = 10*0.1*0.9^9

P[ X = 1 ] = 0.3874

P[ X = 2 ] = 10C2*0.1^2*(1-0.1)^(10-2)

P[ X = 2 ] = 10C2*0.1^2*0.9^8

P[ X = 2 ] = 45*0.1^2*0.9^8

P[ X = 2 ] = 0.1937

P[ X <= 2 ] = 0.3487 + 0.3874+ 0.1937

P[ X <= 2 ] = 0.9298

Probability that it will be flooded at most twice during a 10-year period = 0.9298