Sunday, July 26, 2020

Probability Part 4 - Practice Problems and Answers (#18)

Problems:

In a lottery game, a player picks six numbers from 1 to 48.  If 5 of the 6 numbers match those drawn, the player wins second prize.  What is the probability of winning this prize?

Compute the probability that a 5-card poker hand is dealt to you that contains four aces?

A company estimates that 0.7% of their products will fail after the original warranty period but within two years of the purchase, with a replacement cost of $350.  If they offer a 2 year extended warranty for $48, what is the company’s expected value of each warranty sold?

Answers:

In a lottery game, a player picks six numbers from 1 to 48. If 5 of the 6 numbers match those drawn, the player wins second prize. What is the probability of winning this prize? Answer: The number of possible outcomes is 48 choose 6 which is equal to 12, 271, 512. The number of ways to choose 5 out of the 6 winning numbers is 6 choose 5 which equals 6. The number of ways to choose 1 out of the 42 of the losing numbers is 42 choose 1 which is 42. By the basic counting rule, the number of favorable outcomes is: (6)(42) = 252. So, the probability of winning this prize is 252/12,271,512 = 21/1,022,626 which is approximately 0.0000205.

Compute the probability that a 5-card poker hand is dealt to you that contains four aces. Answer: The number of possible outcomes is 52 choose 5 which is equal to 2,598,960. Since there are four aces, there are 4 choose 4 equals 1 way to select four aces. Since there are 48 non-aces and we want one of them, there are 48 choose 1 = 48 ways to select one of the non-aces. By the Basic Counting Rule, there are (1)(48) = 48 ways to choose four aces and one non-ace. Thus, the probability of four aces is 48/2,598,960 = 1/54,145 which is approximately 0.0000185.

A company estimates that 0.7% of their products will fail after the original warranty period but within two years of the purchase, with a replacement cost of $350. If they offer a 2 year extended warranty for $48, what is the company’s expected value of each warranty sold? Answer: The outcome $48 - $350 = -$302 has a probability of 0.007. The outcome $48 has a probability of 0.993. The expected value is (-$302)(0.007) + ($48)(0.993) = $45.55.

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