A person owns both her own business and her own home and purchases a cariety of insurances to protect them. In any one year the probability of an insurance claim on the business is .4 and on the home is .05. Assuming that these are independent events, find the probability that this person will have an insurance claim at
a.) both home and business
b.) exactly one place
c.) neither
a: Pr= .4*.05
b: Pr=.4*.95+ .05*.6=..
c: Pr= 1-Pr(a)-Pr(b)
To find the probabilities in this scenario, we can use the principles of probability and the fact that the events (insurance claims on the business and home) are assumed to be independent.
a.) To find the probability that the person will have an insurance claim at both home and business, we multiply the individual probabilities of each event. In this case, the probability of an insurance claim on the business is 0.4, and on the home is 0.05. Therefore, the probability of having an insurance claim at both places is:
P(claim at home and business) = P(claim at home) × P(claim at business) = 0.05 × 0.4 = 0.02 or 2%.
b.) To find the probability of having an insurance claim at exactly one place (either home or business), there are two possibilities: claim at home and no claim at business, or claim at business and no claim at home. We can calculate this by using the complement rule, which states that the probability of event A occurring is equal to 1 minus the probability of event A not occurring. Therefore, the probability of having an insurance claim at exactly one place is:
P(exactly one claim) = P(claim at home) × P(no claim at business) + P(no claim at home) × P(claim at business)
= 0.05 × (1 - 0.4) + (1 - 0.05) × 0.4
= 0.05 × 0.6 + 0.95 × 0.4
= 0.03 + 0.38
= 0.41 or 41%.
c.) To find the probability of having neither insurance claim, we calculate the complement of having at least one claim. So, the probability of having neither claim is:
P(neither claim) = 1 - P(at least one claim)
= 1 - (P(claim at home) × P(no claim at business) + P(no claim at home) × P(claim at business))
= 1 - (0.05 × 0.6 + 0.95 × 0.4)
= 1 - (0.03 + 0.38)
= 1 - 0.41
= 0.59 or 59%.
In summary, the probabilities are:
a.) 2% chance of having an insurance claim at both home and business.
b.) 41% chance of having an insurance claim at exactly one place.
c.) 59% chance of having neither insurance claim.