Questions: In a deck of cards, there are 52 cards split evenly between red suits (hearts and diamonds) and black suits (clubs and spades). In each suit, there are cards numbered 2 through 10, a jack, a queen, a king and an ace. What is the probability of selecting two spades in a row? a) 7.2 % b) 6.5 % c) 6.3 % d) 5.9 % e) 5.8 %

Solution Steps

To find the probability of selecting two spades in a row from a deck of 52 cards, we first calculate the probability of drawing a spade on the first draw. Then, assuming a spade was drawn, we calculate the probability of drawing another spade on the second draw. Multiply these probabilities to get the final probability.

The probability of drawing a spade on the first draw is given by the ratio of the number of spades to the total number of cards. There are 13 spades in a deck of 52 cards, so the probability is:

\[ P(\text{first spade}) = \frac{13}{52} = \frac{1}{4} = 0.25 \]

Assuming the first card drawn is a spade, there are now 12 spades left in a deck of 51 cards. The probability of drawing a spade on the second draw is:

\[ P(\text{second spade} \mid \text{first spade}) = \frac{12}{51} \approx 0.2353 \]

The probability of both events (drawing two spades in a row) is the product of the individual probabilities:

\[ P(\text{two spades}) = P(\text{first spade}) \times P(\text{second spade} \mid \text{first spade}) = 0.25 \times 0.2353 \approx 0.05882 \]

To express the probability as a percentage, multiply by 100:

\[ P(\text{two spades}) \times 100 \approx 5.882\% \]

Final Answer