A random walker starts at 0 on the [math]x[/math]-axis and at each time unit moves 1 step to the right or 1 step to the left with probability 1/2. Estimate the probability that, after 100 steps, the walker is more than 10 steps from the starting position.

• 0.23
• 0.26
• 0.29
• 0.32
• 0.35

References

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

A piece of rope is made up of 100 strands. Assume that the breaking strength of the rope is the sum of the breaking strengths of the individual strands. Assume further that this sum may be considered to be the sum of an independent trials process with 100 experiments each having expected value of 10 pounds and standard deviation 1. Find the approximate probability that the rope will support a weight of 985 pounds.

• 0.9
• 0.93
• 0.95
• 0.97
• 0.99

References

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

A student takes a multiple-choice test with 40 questions. The probability that the student answers a given question correctly is 0.5, independent of all other questions. The probability that the student answers more than [math]N[/math] questions correctly is greater than 0.10. The probability that the student answers more than [math]N+1[/math] questions correctly is less than 0.10.

Calculate [math]N[/math] using a normal approximation with the continuity correction.

• 23
• 25
• 32
• 33
• 35

A city has just added 100 new female recruits to its police force. The city will provide a pension to each new hire who remains with the force until retirement. In addition, if the new hire is married at the time of her retirement, a second pension will be provided for her husband. A consulting actuary makes the following assumptions:

1. Each new recruit has a 0.4 probability of remaining with the police force until retirement.
2. Given that a new recruit reaches retirement with the police force, the probability that she is not married at the time of retirement is 0.25.
3. The events of different new hires reaching retirement and the events of different new hires being married at retirement are all mutually independent events.

Calculate the probability that the city will provide at most 90 pensions to the 100 new hires and their husbands.

• 0.60
• 0.67
• 0.75
• 0.93
• 0.99

Let [math]X[/math] and [math]Y[/math] be the number of hours that a randomly selected person watches movies and sporting events, respectively, during a three-month period. The following information is known about [math]X[/math] and [math]Y[/math]:

• [math]\operatorname{E}(X) = 50 [/math]
• [math]\operatorname{E}(Y) = 20 [/math]
• [math]\operatorname{Var}(X) = 50 [/math]
• [math]\operatorname{Var}(Y) = 30 [/math]
• [math]\operatorname{Cov}(X,Y) = 10 [/math]

The totals of hours that different individuals watch movies and sporting events during the three months are mutually independent.

One hundred people are randomly selected and observed for these three months. Let [math]T[/math] be the total number of hours that these one hundred people watch movies or sporting events during this three-month period.

Approximate the value of [math]\operatorname{P}[T \lt 7100][/math].

• 0.62
• 0.84
• 0.87
• 0.92
• 0.97

The total claim amount for a health insurance policy follows a distribution with density function

The premium for the policy is set at the expected total claim amount plus 100. If 100 policies are sold, calculate the approximate probability that the insurance company will have claims exceeding the premiums collected.

• 0.001
• 0.159
• 0.333
• 0.407
• 0.460

Find the probability that among 10,00 random digits the digit 3 appears not more than 931 times.

• 0.9
• 0.925
• 0.95
• 0.975
• 0.99

References

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

A balanced coin is flipped 400 times. Determine the number [math]x[/math] such that the probability that the number of heads is between [math]200- x[/math] and [math]200 + x[/math] is approximately .80.

• 7
• 9
• 11
• 13
• 15

References

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

A bank accepts rolls of pennies and gives 50 cents credit to a customer without counting the contents. Assume that a roll contains 49 pennies 30 percent of the time, 50 pennies 60 percent of the time, and 51 pennies 10 percent of the time. How many rolls does the bank need to collect to have a 99 percent chance of a net loss?

• 20
• 24
• 35
• 45
• 49

References

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

A surveying instrument makes an error of [math]-2[/math], [math]-1[/math], 0, 1, or 2 feet with equal probabilities when measuring the height of a 200-foot tower. Estimate the probability that in 18 independent measurements of this tower, the average of the measurements is between 199 and 201, inclusive.

• 0.975
• 0.98
• 0.983
• 0.99
• 0.997

References

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