Three cards are drawn without replacement from an ordinary deck of 52 playing cards. What is the probability that the second and third cards are spades if the first card was not a spade?
Answer: 36/52 18/26 9/13= 69%.
Is this right? Thanks for the help.
Wrong question please don't answer. Thanks.
No, that is not right. Doesn't 69% seem too high a probability for drawing spades twice in a row?
This is a conditional probability. A key phrase is "...IF the first card was not a spade." You have to assume that the first card was NOT a spade. Then the deck before the second draw has 13 spades out of 52. The same applies to the third draw, since the drawn spades are replaced. The answer is (13/52)^2 = 1/16
If we consider "without replacement", the second draw starts with only 51 cards and 13 spades.
The probability of success would be the product of the probabilities of the second and third draws, namely
(13/51)*(12/50)=26/425
I did the with replacement case by mistake. Careless of me. Sorry
To find the probability that the second and third cards are spades, given that the first card was not a spade, we can break down the problem step by step.
First, we need to determine the number of spades and non-spades in the deck. There are 13 spades and 39 non-spades in a standard deck of 52 cards.
Since the first card drawn was not a spade, there are now 12 spades and 39 non-spades remaining in the deck for the second card draw.
The probability of drawing a spade for the second card is calculated by taking the number of favorable outcomes (12 spades) divided by the number of total possible outcomes (51 cards remaining in the deck). So, the probability of drawing a spade for the second card is 12/51.
After the second card is drawn, there are now 11 spades and 38 non-spades remaining in the deck for the third card draw.
Similarly, the probability of drawing a spade for the third card is 11/50.
To find the probability that both the second and third cards are spades, we multiply the probabilities of the individual events together. Therefore, the probability is (12/51) * (11/50) = 132/2550.
This fraction can be simplified by dividing both numerator and denominator by their greatest common divisor, which in this case is 6. So, the simplified fraction is 22/425.
To express this probability as a percentage, we divide the numerator by the denominator, giving 22/425 ≈ 5.2%. Therefore, the probability that the second and third cards are spades, given that the first card was not a spade, is approximately 5.2%.
Hence, the answer of 36/52 or 69% provided is incorrect. The correct probability is approximately 5.2%.