Question

An automobile insurance company predicts that 15% of policy holders file a claim within the next year. Since this company has many policy holders, we may assume that filing a claim is independent between policy holders.

a. If 15 policy holders are randomly selected, what is the probability that exactly two will file a claim within the next year? What calculator command and inputs did you use?

b. If 15 policy holders are randomly selected, what is the probability that less than four will file a claim within the next year? What calculator command and inputs did you use?

Answer 1

# Let X be a random variable denoting whether a polyciyholder files a claim within the next year or not.
# There are only two possible outcomes: "files a claim" and "does not file a claim"
# Thus, it becomes a Bernoulli random variable taking value 1 (if files a claim) and 0 (otherwise) with probability of success 15% (i.e. p=0.15).
# X ~ Bernoulli(p)
# Here, there are 15 policy holders, thus, n=15 independent Bernoulli random variables.
# Let Y=sum(X) be another random variable that denotes the number of holders filing the claim within the next year.
# Y ~ Binomial(n,p) where, n=15, p=0.15

### the probability that exactly two will file a claim within the next year is P(Y=2)
### the probability that less than four will file a claim within the next year is P(Y<4)=P(Y<=3)

USED R-SOFTWARE:
# R-commands and outputs:
## "dbinom()" command gives the probability mass function value at a point.
## "pbinom()" command gives the cumulative probability mass function value at a point

p=15/100 #15%
n=15
dbinom(2,n,p) #P(Y=2)
### [1] 0.2856392

x=0:3 # less than 4
pbinom(3,n,p) #P(Y<4)
### [1] 0.8226552

(Any) Statistics Calculator: