Three cards are drawn in succession (without replacement) from a 52-card deck. Find the probability that: If the first is an ace, then the other two are aces.
I know this has to do with conditional probability, like "What is the probability that the second and third cards picked are aces given that the first card picked is definitely an ace".
What I did was just multiply (3/51) x (2/50). Is this correct?
No. You would multiply 1/52 by 1/51 by 1/50. This is because the first time you picked 1 card out of 52. Next you picked 1 card out of 51. Next you picked 1 card out of 50. So you multiply all three fractions and get. 1/132600. This is probably not true, but, as my name says, Im Not So Smart.
Hannah. I believe you are correct. The answer by Im Not So Smart is clearly wrong.
Yes, you are correct. To calculate the probability that the second and third cards picked are aces given that the first card picked is definitely an ace, you can multiply the probabilities of each event occurring.
The probability of picking an ace as the first card is 4/52, since there are 4 aces in a 52-card deck. After the first ace is picked, there are 3 aces left in a deck of 51 cards, so the probability of picking an ace as the second card is 3/51. Finally, after the first and second aces are picked, there are 2 aces left in a deck of 50 cards, so the probability of picking an ace as the third card is 2/50.
Therefore, the probability that the second and third cards picked are aces given that the first card picked is definitely an ace is (3/51) × (2/50) = 6/2550 = 1/425.
To find the probability that the other two cards are aces given that the first card is an ace, you need to consider the fact that the cards are drawn without replacement. This means that the number of cards decreases with each draw.
Here's how you can solve this problem step by step:
Step 1: Determine the probability of drawing an ace as the first card.
There are four aces in a deck, so the probability of drawing an ace as the first card is 4/52.
Step 2: Determine the probability of drawing an ace as the second card, given that the first card is an ace.
After drawing the first ace, there are now only 51 cards left in the deck. Since we want the second card to be an ace, there are only 3 aces remaining. So, the probability of drawing an ace as the second card, given that the first card is an ace, is 3/51.
Step 3: Determine the probability of drawing an ace as the third card, given that the first two cards are aces.
After drawing the first two aces, there are only 50 cards left in the deck. Out of these, there are only 2 aces remaining. So, the probability of drawing an ace as the third card, given that the first two cards are aces, is 2/50.
Step 4: Multiply the probabilities together.
To find the overall probability, you need to multiply the probabilities from each step together:
(4/52) x (3/51) x (2/50) = 24/132600 = 1/5525
So, the probability that the other two cards are aces, given that the first card is an ace, is 1/5525.
Therefore, your approach was correct, and the answer is indeed (3/51) x (2/50).