Question

Amy and samiha have a hat that contains two playing cards, one ace and one king. they are playing a game where they randomly select a card out of the hat four times, with replacement. samiha thinks that the probability of getting either an ace or a king on the first pick is greater than the probability of getting at least one ace in the four picks. amy disagrees. she thinks that the two probabilities are equal. the sample space of possible outcomes is listed below. a represents an ace, and k represents a king. who is correct, amy or samiha?

a. Amy is correct
b. Samiha is correct

Answer 1

Amy is correct. The probability of getting at least one ace in the four picks is greater than the probability of getting either an ace or a king on the first pick.

Step-by-step explanation:

Amy and Samiha are playing a game where they randomly select a card out of a hat containing an ace and a king, four times with replacement. Samiha thinks that the probability of getting either an ace or a king on the first pick is greater than the probability of getting at least one ace in the four picks. Amy disagrees and thinks the two probabilities are equal. Let's calculate the probabilities to determine who is correct.

Probability of getting either an ace or a king on the first pick:

There are two favorable outcomes (ace or king) out of the two possible outcomes (ace, king), so the probability is 2/2 = 1.

Probability of getting at least one ace in the four picks:

The probability of not getting an ace on one pick is 1 - (1/2) = 1/2. Since the picks are independent, the probability of not getting an ace in four picks is (1/2)^4 = 1/16. Therefore, the probability of getting at least one ace in four picks is 1 - 1/16 = 15/16.

Based on the calculations, Amy is correct. The probability of getting at least one ace in the four picks is greater than the probability of getting either an ace or a king on the first pick.