Question

In a card game, the probability that you will have a hand with three of a kind is about 6%. The dealer wants to know the probability for a player to be dealt three of a kind by at least the second hand. Should a geometric probability density function or a cumulative distribution function be used? Explain.

Answer 1

The cumulative distribution function is the appropriate method to calculate the probability of a player being dealt three of a kind at least by the second hand in a card game, which involves summing the probabilities of the event occurring on the first and second hand.

Step-by-step explanation:

To determine the probability of a player being dealt three of a kind at least by the second hand in a card game, we should use a cumulative distribution function. This function will sum the probabilities of the event occurring in the first hand and the probability of it not occurring in the first hand but occurring in the second. Since the event ('three of a kind') is independent between hands, we can use the complement rule and multiplication rule of probability.

The formula to use is:

1. Compute the probability of getting three of a kind on the first hand: P(1st hand) = 0.06 (given).
2. Compute the probability of not getting three of a kind on the first hand and then getting it on the second: P(2nd hand) = P(not on 1st hand) * P(on 2nd hand) = (1 - P(1st hand)) * P(1st hand) = (1 - 0.06) * 0.06.
3. Sum these two probabilities to get the cumulative probability: P(at least by 2nd hand) = P(1st hand) + P(2nd hand).

The use of a geometric probability density function would not be appropriate because it is used for modeling the number of trials until the first success, not for calculating cumulative probabilities across multiple trials. To solve the dealer's query, a cumulative distribution function is the correct tool.