Which is more likely event a drawing an ace, then a king, then a queen or event to be drawn, three consecutive aces ? Assume that card drawn or not return to the pack how does your answer change if Carter return after each draw explain how a determines your answers.
To compare the likelihood of two events, we need to calculate their probabilities.
Event A: Drawing an ace, then a king, then a queen.
Event B: Drawing three consecutive aces.
Each event consists of drawing cards from a standard deck of 52 cards. Without replacement means that the drawn card is not returned to the deck.
Event A:
The probability of drawing an ace is 4/52 since there are four aces in a deck of 52 cards.
The probability of drawing a king, given that an ace has already been drawn, is 4/51 since there are four kings remaining in a deck of 51 cards.
The probability of drawing a queen, given that an ace and a king have already been drawn, is 4/50 since there are four queens remaining in a deck of 50 cards.
The probability of event A occurring is calculated by multiplying these individual probabilities:
P(A) = (4/52) * (4/51) * (4/50) = 64/16575
Event B:
The probability of drawing an ace is 4/52 since there are four aces in a deck of 52 cards.
The probability of drawing another ace, given that one ace has already been drawn, is 3/51 since there are three aces remaining in a deck of 51 cards.
The probability of drawing a third ace, given that two aces have already been drawn, is 2/50 since there are two aces remaining in a deck of 50 cards.
The probability of event B occurring is calculated by multiplying these individual probabilities:
P(B) = (4/52) * (3/51) * (2/50) = 24/132600
Comparing the probabilities, we can see that:
P(A) = 64/16575 ≈ 0.00386
P(B) = 24/132600 ≈ 0.00018
Therefore, event A is more likely to occur than event B.
If the cards were returned to the deck after each draw, meaning they were replaced, the calculations would change:
Event A:
P(A) = (4/52) * (4/52) * (4/52) ≈ 0.002641
Event B:
P(B) = (4/52) * (4/52) * (4/52) ≈ 0.002641
In this case, both events have the same probability, as each draw is independent and does not affect the deck's composition.
To determine which event is more likely, we need to calculate the probabilities of each event.
Event A: Drawing an ace, then a king, then a queen
If the cards are drawn without replacement (i.e., cards are not returned to the pack), the probability of drawing an ace first is 4/52 (there are 4 aces in a standard deck of 52 cards). Once the ace is drawn, there are now 51 cards left, so the probability of drawing a king is 4/51. Finally, after drawing the king, there are 50 cards left, so the probability of drawing a queen is 4/50.
The probability of drawing an ace, then a king, then a queen is (4/52) x (4/51) x (4/50).
Event B: Drawing three consecutive aces
Again, assuming cards are not returned to the pack, the probability of drawing an ace first is 4/52. After the first ace is drawn, there are 51 cards left, and the probability of drawing another ace is 3/51. Finally, after the second ace is drawn, there are 50 cards left, and the probability of drawing the third and final ace is 2/50.
The probability of drawing three consecutive aces is (4/52) x (3/51) x (2/50).
If cards are returned to the pack after each draw, then the probability of drawing an ace, king, or queen on each subsequent draw remains the same for each event. The only difference is that the denominator for each probability calculation would remain 52, since the total number of cards in the deck does not change.
Comparing the two events:
- Without returning cards to the pack, the probability of event A is (4/52) x (4/51) x (4/50).
- Without returning cards to the pack, the probability of event B is (4/52) x (3/51) x (2/50).
- With returning cards to the pack, the probabilities of events A and B remain the same.
Comparing the exact probabilities will require calculations, but generally speaking, event A (drawing an ace, then a king, then a queen) is more likely than event B (drawing three consecutive aces) since it involves drawing three cards from a total of 52, rather than drawing three specific cards consecutively.