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Probability and Expected Value Calculations

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Probability

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Make sure that you state your answer in a way that does not lose precision. Show all digits.

a. The Personal Identification Number (PIN) for your ATM card is three places long. Any number is permitted in each place. How many possible values of a PIN are there?

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b. Your bank has decided to allow English alphabetic characters (upper and lower case are permitted) as well as any numeric digit in your PIN. How many possible PINs are there when that change is made?

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c. In addition to b. above, your bank has decided to require that the first place of the PIN be an English upper case alphabetic character. How many possible PINs are there when that additional change is made?

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d. In addition to b. and c. above, your bank has changed the PIN size to four places. How many possible PINs are there when that change is made?

Theoretical Probability â€“ show each as a fraction

a. What is the theoretical probability of rolling one die and getting a 2 ?

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b. What is the theoretical probability of rolling one die and getting a 4 or a 5?

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c. What is the theoretical probability of rolling one die and not getting a 5?

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d. What is the theoretical probability of rolling one die and getting a 7 ?

Theoretical Probability (and)

Â What is the theoretical probability that your five best friends all have telephone numbers ending in 5?Â  (Hint: first determine the probability of a single telephone number ending with a 5.Â  Then calculate the probability of the five independent events occurring together.) Show your answer as a fraction.

Empirical ProbabilityÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

After recording the forecasts of your local weather predictor for 125 days, you conclude that he/she gave a correct forecast 84 times.Â  Calculate the empirical probability of correct forecasts during that time. Show your answer as a fraction.

Probability calculations

Â Studies of Florida weather show that, historically, the Miami region is hit by a hurricane every 25 years. Calculate the following probabilities based on the historical record.

a. What is the probability that Miami will be hit by a hurricane in any given year? Show as a fraction.

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b. What is the probability that Miami will be hit by hurricanes in both of the next two consecutive years? Show as a fraction.

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c. What is the probability that Miami will be hit by hurricanes in either the next year or the year after? Show as a fraction.

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d. What is the probability that Miami will be hit by at least one hurricane in the next seven years? Show as a percentage with three decimal places.

Probability calculations â€“ exactly one occurrence

You are looking for a job. Eventually you get six interviews. In each case you are one of four final candidates for the respective job. Let us assume you have a .25 probability of getting each job and the probability of getting each job offer is independent of getting any other job offer.

a.. What is the probability that you get exactly one job offer of the six opportunities?

Show this as a percentage with three decimal places.

Expected Value â€“ seller

An insurance company sells a policy for \$1500.Â

Based on past data,

an average of 1 in 100 policyholders will win a \$25,000 claim on a policy,

an average of 1 in 200 policyholders will win a \$50,000 claim on a policy, and

an average of 1 in 500 policyholders will win a \$250,000 claim on a policy.Â

a. Find the expected value (to the company) per policy sold.

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b. If the company sells 10,000 policies, what is the expected total profit or loss for the company?

You can buy a lottery ticket for \$500.Â

The odds are as follows:

1 in 10 tickets will pay \$500.

1 in 100 tickets will pay \$1000.

1 in 1000 tickets will pay \$200,000.

1 in 10,000 tickets will pay \$1,000,000.Â

1 in 100,000 tickets will pay \$10,000,000.Â

a. What is the expected value of the ticket to you?

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