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# Discrete Distributions probability problems – SOA Exam P CAS Exam 1

## Binomial Distribution Bayes Rule example question

A hospital receives  of its flu vaccine shipments from Company X and the remainder of its shipments from other companies. Each shipment contains a very large number of vaccine vials. For Company X’s shipments, 10% of the vials are ineffective. For every other company, 2% of the vials are ineffective. The hospital tests 30 randomly selected vials from a shipment and finds that one vial is ineffective. What is the probability that this shipment came from Company X?

Solution to this Society of Actuaries Exam P practice problem is given in the video below!

## Binomial Distribution example problem

A company prices its hurricane insurance using the following assumptions:

(i) In any calendar year, there can be at most one hurricane

(ii) In any calendar year, the probability of a hurricane is 0.05

(iii) The number of hurricanes in any calendar year is independent of the number of hurricanes in any other calendar year.

Using the company’s assumptions, calculate the probability that there are fewer than 3 hurricanes in a 20-year period.

Solution to this Society of Actuaries Exam P practice problem is given in the video below!

## Binomial Distribution Hard example

A company establishes a fund of 120 from which it wants to pay an amount, C, to any of its 20 employees who achieve a high performance level during the coming year. Each employee has a 2% chance of achieving a high performance level during the coming year, independent of any other employee. Determine the maximum value of C for which the probability is less than 1% that the fund will be inadequate to cover all payments for high performance.

Solution to this Society of Actuaries Exam P practice problem is given in the video below!

## Binomial Distribution Multiple Random Variables example question

A study is being conducted in which the health of two independent groups of ten policyholders is being monitored over a one-year period of time. Individual participants in the study group drop out before the end of the study with probability 0.2 (independently of the other participants). What is the probability that at least 9 participants complete the study in one of the two groups, but not in both groups?

Solution to this Society of Actuaries Exam P practice problem is given in the video below!

## Poisson Distribution example problem

An actuary has discovered that policyholders are three times as likely to file two claims as to file four claims. If the number of claims filed has a Poisson distribution, what is the variance of the number of claims filed?

Solution to this Society of Actuaries Exam P practice problem is given in the video below!

1. Brent says: