three cards are drawn without replacement from an ordinary deck of 52 playing cards. What is the pronanility that the third card is a spade if the first two cards were not spades?
How do I even start this?
remaining is 13 spades out of 50 cards.
Pr=13/50
To solve this problem, we need to break it down into smaller steps.
Step 1: Calculate the probability of drawing a non-spade as the first card.
Since the first card can be any card from the 52-card deck, the probability of drawing a non-spade as the first card is given by:
P(first card is a non-spade) = number of non-spades / total number of cards in the deck
There are 13 non-spade cards (clubs, diamonds, and hearts) in the deck, so the probability is:
P(first card is a non-spade) = 13 / 52 = 1/4
Step 2: Calculate the probability of drawing a non-spade as the second card.
After drawing the first card, the deck now contains 51 cards, including 12 non-spade cards. Hence, the probability of drawing a non-spade as the second card is given by:
P(second card is a non-spade) = number of non-spades / total number of remaining cards in the deck
P(second card is a non-spade) = 12 / 51
Step 3: Calculate the probability of drawing a spade as the third card.
After drawing the first two cards, the remaining deck contains 50 cards, including 13 spade cards. Therefore, the probability of drawing a spade as the third card is given by:
P(third card is a spade) = number of spades / total number of remaining cards in the deck
P(third card is a spade) = 13 / 50
Step 4: Multiply the probabilities together.
The probability that the third card is a spade given that the first two cards were not spades is the product of the probabilities calculated in steps 1, 2, and 3:
P(third card is a spade | first two cards were not spades) = P(first card is a non-spade) * P(second card is a non-spade) * P(third card is a spade)
P(third card is a spade | first two cards were not spades) = (1/4) * (12/51) * (13/50) ≈ 0.061
Therefore, the probability that the third card drawn is a spade, given that the first two cards were not spades, is approximately 0.061 or 6.1%.