Two cards are drawn without replacement from an ordinary deck of 52 playing cards. What is the probability that both are spades if the first card drawn was a spade?
It equals the fraction of remaining cards that are spades AFTER one has already been removed. That would be 3/51 or 1/17.
To find the probability that both cards are spades given that the first card drawn was a spade, we need to use the concept of conditional probability. Let's break down the problem step by step:
Step 1: Determine the number of spades in the deck.
Since we are working with an ordinary deck of 52 playing cards, there are 13 spades in the deck.
Step 2: Calculate the probability of drawing a spade on the first card.
Since the first card drawn was specified to be a spade, the probability of this occurring is simply 13/52, or 1/4.
Step 3: Determine the number of remaining spades in the deck.
Since we have drawn one spade already, there are now 12 spades remaining in the deck.
Step 4: Calculate the probability of drawing a spade on the second card given that the first card was a spade.
Since we are drawing without replacement, the total number of cards in the deck decreased to 51. Therefore, the probability of drawing a spade on the second card is 12/51.
Step 5: Calculate the conditional probability of drawing two spades.
To calculate the probability of both cards being spades given that the first card was a spade, we multiply the probabilities from steps 2 and 4:
P(both spades|first card spade) = (1/4) * (12/51)
Step 6: Simplify the fraction (optional).
If desired, we can simplify this fraction. In this case, both the numerator and denominator are divisible by 3, so we get:
P(both spades|first card spade) = (1/4) * (4/17) = 1/17
Therefore, the probability that both cards drawn are spades, given that the first card drawn was a spade, is 1/17.