The number of policies that an agent sells has a Poisson distribution with modes at 2 and 3.

[math]K[/math] is the smallest number such that the probability of selling more than [math]K[/math] policies is less than 25%.

Calculate [math]K[/math].

• 1
• 2
• 3
• 4
• 5

The number of traffic accidents occurring on any given day in Coralville is Poisson distributed with mean 5. The probability that any such accident involves an uninsured driver is 0.25, independent of all other such accidents.

Calculate the probability that on a given day in Coralville there are no traffic accidents that involve an uninsured driver.

• 0.007
• 0.010
• 0.124
• 0.237
• 0.287

A group of 100 patients is tested, one patient at a time, for three risk factors for a certain disease until either all patients have been tested or a patient tests positive for more than one of these three risk factors. For each risk factor, a patient tests positive with probability [math]p[/math], where [math]0 \lt p \lt 1[/math]. The outcomes of the tests across all patients and all risk factors are mutually independent.

Determine an expression for the probability that exactly [math]n[/math] patients are tested, where [math]n[/math] is a positive integer less than 100.

• [math][1-3p^2(1-p)]^{n-1}[3p^2(1-p)][/math]
• [math][1-3p^2(1-p) - p^3]^{n-1}[3p^2(1-p) + p^3][/math]
• [math][1-3p^2(1-p) - p^3]^{n-1}[3p^2(1-p) + p^3]^{n-1}[/math]
• [math]n[1-3p^2(1-p) - p^3]^{n-1}[3p^2(1-p) + p^3][/math]
• [math]3[(1 − p )^{n −1} p][1-(1-p)^{n-1}p] + [(1-p)^{n-1}p]^3[/math]

A flood insurance company determines that [math]N[/math], the number of claims received in a month, is a random variable with [math]\operatorname{P}[N=n] = \frac{2}{3^{n+1}}[/math] for [math]n = 1,2, \ldots [/math]. The numbers of claims received in different months are mutually independent.

Calculate the probability that more than three claims will be received during a consecutive two-month period, given that fewer than two claims were received in the first of the two months.

• 0.0062
• 0.0123
• 0.0139
• 0.0165
• 0.0185

1. [math]X[/math] and [math]Y[/math] are Poisson distributed.
2. The first moment of [math]X[/math] is less than the first moment of [math]Y[/math] by 8.
3. The second moment of [math]X[/math] is 60% of the second moment of [math]Y[/math].

Calculate the variance of [math]Y[/math].

• 4
• 12
• 16
• 27
• 35

A company has purchased a policy that will compensate for the loss of revenue due to severe weather events. The policy pays 1000 for each severe weather event in a year after the first two such events in that year. The number of severe weather events per year has a Poisson distribution with mean 1.

Calculate the expected amount paid to this company in one year.

• 80
• 104
• 368
• 512
• 632

The king's coinmaster boxes his coins 500 to a box and puts 1 counterfeit coin in each box. The king is suspicious, but, instead of testing all the coins in 1 box, he tests 2 coins at random from each of 250 boxes. What is the probability that he finds at least one fake?

• 0.60
• 0.63
• 0.66
• 0.69
• 0.72

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

Suppose we have an urn containing 5 yellow balls and 7 green balls. We draw 3 balls, without replacement, from the urn. Find the expected number of yellow balls drawn.

• 1
• 1.1
• 1.15
• 1.2
• 1.25

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

A large portfolio of travel insurance policies contains policies with three different deductible levels: 50% have a deductible of $200, 25% have a deductible of $400 and 25% have a deductible of $600. Ten policies are randomly chosen from the portfolio of policies two of which have a deductible of $600. Determine the probability that more than six of the policies chosen have a deductible of $400.

1. 0
2. 0.0351
3. 0.1951
4. 0.225
5. 0.2351

The claim frequency [math]N[/math] has the following properties:

• The claim frequency has a geometric distribution
• The probability that [math]N=1[/math] given that [math]N\gt0[/math] equals 0.3

Determine the variance of the claim frequency given that [math]N\gt0[/math].

1. 0.4671
2. 1.19
3. 3.1477
4. 3.477
5. 7.77