A die is rolled 24 times. Use the Central Limit Theorem to estimate the probability that

• the sum is greater than 84.
• the sum is equal to 84.

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.

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 1000 pounds.
• of 970 pounds.

Write a program to find the average of 1000 random digits 0, 1, 2, 3, 4,5, 6, 7, 8, or 9. Have the program test to see if the average lies within three standard deviations of the expected value of 4.5. Modify the program so that it repeats this simulation 1000 times and keeps track of the number of times the test is passed. Does your outcome agree with the Central Limit Theorem?

A die is thrown until the first time the total sum of the face values of the die is 700 or greater. Estimate the probability that, for this to happen,

• more than 210 tosses are required.
• less than 190 tosses are required.
• between 180 and 210 tosses, inclusive, are required.

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.

• Find the expected value and the variance for the amount that the bank loses on a typical roll.
• Estimate the probability that the bank will lose more than 25 cents in 100 rolls.
• Estimate the probability that the bank will lose exactly 25 cents in 100 rolls.
• Estimate the probability that the bank will lose any money in 100 rolls.
• How many rolls does the bank need to collect to have a 99 percent chance of a net loss?

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.

• Find the expected value and the variance for the height obtained using this instrument once.
• Estimate the probability that in 18 independent measurements of this tower, the average of the measurements is between 199 and 201, inclusive.

For Example estimate [math]P(S_{30} = 0)[/math]. That is, estimate the probability that the errors cancel out and the student's grade point average is correct.

Prove the Law of Large Numbers using the Central Limit Theorem.

Peter and Paul match pennies 10,00 times. Describe briefly what each of the following theorems tells you about Peter's fortune.

• The Law of Large Numbers.
• The Central Limit Theorem.