Question

1)                 15 policyholders over 50 are selected at random; what is the probability that at least 10 will submit a claim during the year? What is the probability that 4 will submit a claim during the year? How many do you expect to submit a claim. What is the standard deviation?

Answer 1

The number of policyholders that submit a claim can be described as binomial distribution X = Bin(15, 0.3)

The probability that at least 10 will submit a claim during the year:

$P(X≥10) = P(X=10)+...+P(X=15) = 0.003+0.0006+0.0001+0+0+0=$

$= 0.0037$ (The last three probabilities is very small and rounded to 4 digits are equal to 0)

The probability that 4 will submit a claim during the year:

$P(X=4)=0.2186$

How many do you expect to submit a claim:

The expected value of Bin(15, 0.3) is the answer

$E(Bin(15, 0.3) = 15*0.3 = 4.5$

If round, it can be assumed that 4 is the most expected value(P(X=4) is a bit more than P(X=5))

What is the standard deviation:

$D(Bin(15, 0.3)=15*0.3*0.7=3.15$

Standard deviation is equal to $\sqrt{D} =\sqrt{3.15}=1.77$