Two cards are drawn without replacement from a well-shuffled deck of 52 playing cards.
a. What is the probability that the first card drawn is a heart?
b. What is the probability that the second card drawn is a heart if the first card drawn was not a heart?
c.What is the probability that the second card drawn is a heart if the first card drawn was a heart?
I know that there are 13 hearts total.
a. P(1st heart) = 13/52
b. P( 2nd heart l 1st heartcomplement) = 13/51. I guess if one card was drawn on the first attempt, there are 51 remaining but there are still 12 hearts.
c. This one I am a bit unsure of. P(2nd heart l 1st heart) = 13/52 * 12/51?
For A, you're right
For B, you're right
For C, it would just be 12/52 because you've only taken 1/52 away and that was a heart.
Disagree with AJL on C. If you have already drawn a heart and not replaced it, the probability of a second heart is 12/51.
a. You are correct. The probability that the first card drawn is a heart is 13/52 or 1/4.
b. To find the probability that the second card drawn is a heart given that the first card drawn was not a heart, you are correct in thinking that there are 51 cards remaining and 12 hearts remaining.
So the probability would be P(2nd heart | 1st heart complement) = 12/51.
c. To find the probability that the second card drawn is a heart given that the first card drawn was a heart, you would use the conditional probability formula.
P(2nd heart | 1st heart) = (P(1st heart and 2nd heart)) / P(1st heart)
The probability of drawing the first heart is 13/52.
After removing the first heart from the deck, there are 51 cards remaining, with 12 hearts remaining.
So the probability of drawing the second heart after the first heart is 12/51.
Therefore, P(2nd heart | 1st heart) = (13/52) * (12/51) = 1/4 * 4/17 = 1/17.
So the probability that the second card drawn is a heart given that the first card drawn was a heart is 1/17.
To find the probability that the second card drawn is a heart if the first card drawn was a heart (part c), you are on the right track.
Since the first card drawn is a heart, we have 12 hearts left in the deck (out of 51 cards remaining), as one heart has already been drawn. Therefore, the probability of drawing a heart on the second card given that the first card was a heart can be calculated as:
P(2nd heart l 1st heart) = (12/51)
Hence, the correct probability is 12/51.
Well done on parts a and b!