Question

1. in a simple card game, player a wins a point if either a face card, a red prime number, or a black perfect - square number is drawn from a standard deck. otherwise, player b wins a point. assume aces do not count as ones! a) which player has the advantage in this game? support your answer with calculations and an explanation.

Explanation:

Step1: Calculate number of face - cards

A standard deck has 52 cards. There are 12 face - cards (4 Jacks, 4 Queens, 4 Kings). So the number of face - cards $n_{face}=12$.

Step2: Calculate number of red prime - numbered cards

Prime numbers on cards (2, 3, 5, 7) and red cards are hearts and diamonds. For each suit, there are 4 prime - numbered cards. So number of red prime - numbered cards $n_{red\ prime}=2\times4 = 8$.

Step3: Calculate number of black perfect - square numbered cards

Perfect square numbers on cards are 4 and 9. Black suits are spades and clubs. So number of black perfect - square numbered cards $n_{black\ square}=2\times2=4$.

Step4: Calculate total favorable outcomes

Using the addition principle for non - overlapping events, the total number of favorable outcomes for Player A, $n=n_{face}+n_{red\ prime}+n_{black\ square}=12 + 8+4=24$.

Step5: Calculate probabilities

The probability that Player A wins, $P(A)=\frac{24}{52}=\frac{6}{13}$. The probability that Player B wins, $P(B)=1 - P(A)=1-\frac{6}{13}=\frac{7}{13}$.

Since $\frac{7}{13}>\frac{6}{13}$, Player B has the advantage.

Answer:

Player B has the advantage.