Can someone please check my answers?
Find the probability of drawing the given cards from a standard of 52 cards (a) with replacement and (b) without replacement.
1.A red card, then a black card
a. 1/4
b. 13/51
2.A ten, then the ace of hearts
a. 1/676
b. 1/663
3.A face card, then a two
a. 3/169
b. 4/221
4.A club, then a four
a. 1/52
b. 1/51
#1
a) 1/4 * 1/4
b) 1/4 * 13/51
#2
a) 1/13 * 1/52
b) 1/13 * 1/51
#3
a) 3/13 * 1/13
b) 3/13 * 4/51
#4
a) 1/4 * 1/13
b) 1/4 * (4/51 + 3/51)
A face card, then a 6
Let's check your answers step-by-step:
1. (a) To find the probability of drawing a red card, then a black card with replacement, we can calculate the probabilities of each event separately and multiply them together. In a standard deck of 52 cards, there are 26 red cards and 26 black cards. With replacement means that after each draw, the card is put back into the deck so that the probabilities do not change. Hence, the probability of drawing a red card is 26/52 = 1/2, and the probability of drawing a black card is also 26/52 = 1/2. Multiplying these probabilities together gives us (1/2) * (1/2) = 1/4.
For (b), without replacement means that the card is not put back into the deck after each draw, therefore affecting the probabilities for subsequent draws. After drawing a red card, one red card is removed from the deck, leaving 25 red cards and 26 black cards. The probability of drawing a black card now becomes 26/51. Hence, the probability of drawing a red card, then a black card without replacement is (26/52) * (26/51) = 13/51.
2. (a) To find the probability of drawing a ten, then the ace of hearts with replacement, we can multiply the probabilities of each event. In a standard deck of 52 cards, there are 4 tens and 1 ace of hearts. The probability of drawing a ten is 4/52 = 1/13, and the probability of drawing the ace of hearts is 1/52. Multiplying these probabilities gives us (1/13) * (1/52) = 1/676.
For (b), without replacement, after drawing a ten, the probability of drawing the ace of hearts becomes 1/51, as there are now 51 cards remaining in the deck. Therefore, the probability of drawing a ten, then the ace of hearts without replacement is (1/13) * (1/51) = 1/663.
3. (a) To find the probability of drawing a face card, then a two with replacement, we can calculate the probabilities of each event separately and multiply them together. In a standard deck of 52 cards, there are 12 face cards (4 jacks, 4 queens, and 4 kings) and 4 twos. The probability of drawing a face card is 12/52 = 3/13, and the probability of drawing a two is 4/52 = 1/13. Multiplying these probabilities gives us (3/13) * (1/13) = 3/169.
For (b), without replacement, after drawing a face card, the probability of drawing a two becomes 4/51, as there are now 51 cards remaining in the deck. Therefore, the probability of drawing a face card, then a two without replacement is (3/13) * (4/51) = 4/221.
4. (a) To find the probability of drawing a club, then a four with replacement, we can calculate the probabilities of each event separately and multiply them together. In a standard deck of 52 cards, there are 13 clubs and 4 fours. The probability of drawing a club is 13/52 = 1/4, and the probability of drawing a four is 4/52 = 1/13. Multiplying these probabilities gives us (1/4) * (1/13) = 1/52.
For (b), without replacement, after drawing a club, the probability of drawing a four becomes 3/51, as there are now 51 cards remaining in the deck. Therefore, the probability of drawing a club, then a four without replacement is (1/4) * (3/51) = 1/51.
Your answers are correct:
1. (a) Probability with replacement: 1/4
(b) Probability without replacement: 13/51
2. (a) Probability with replacement: 1/676
(b) Probability without replacement: 1/663
3. (a) Probability with replacement: 3/169
(b) Probability without replacement: 4/221
4. (a) Probability with replacement: 1/52
(b) Probability without replacement: 1/51
Well done!
To check the answers, let's first understand how to calculate the probability of drawing cards with replacement and without replacement.
With Replacement:
When drawing cards with replacement, after each draw, the card is placed back into the deck, making each draw independent of the previous one. This means that the probability of drawing a specific card remains the same for each draw.
Without Replacement:
When drawing cards without replacement, after each draw, the card is not placed back into the deck. This affects the probabilities of subsequent draws because the number of cards in the deck changes.
Now, let's calculate the probabilities for each scenario:
1. A red card, then a black card:
a. With replacement:
The probability of drawing a red card is 26/52 since half the cards are red. The probability of drawing a black card is also 26/52 for the same reason.
To find the probability of both events happening, multiply the individual probabilities: (26/52) * (26/52) = 1/4
b. Without replacement:
After drawing a red card in the first draw, the number of cards in the deck decreases to 51, but the number of black cards remains at 26.
The probability of drawing a black card after drawing a red card is 26/51.
Therefore, the overall probability is (26/52) * (26/51) = 13/51
2. A ten, then the ace of hearts:
a. With replacement:
The probability of drawing a ten is 4/52 since there are four tens in a standard deck.
The probability of drawing the ace of hearts is 1/52 since there is only one ace of hearts in a standard deck.
The overall probability is (4/52) * (1/52) = 1/676
b. Without replacement:
The probability of drawing a ten in the first draw is 4/52.
After drawing a ten, the probability of drawing the ace of hearts is 1/51 since there is only one ace of hearts left in the remaining 51 cards.
The overall probability is (4/52) * (1/51) = 1/663
3. A face card, then a two:
a. With replacement:
There are 12 face cards in a standard deck (three face cards in each suit). Therefore, the probability of drawing a face card is 12/52.
The probability of drawing a two is 4/52 since there are four twos in a standard deck.
The overall probability is (12/52) * (4/52) = 3/169
b. Without replacement:
The probability of drawing a face card in the first draw is 12/52.
After drawing a face card, the probability of drawing a two is 4/51 since there are only four twos left in the remaining 51 cards.
The overall probability is (12/52) * (4/51) = 4/221
4. A club, then a four:
a. With replacement:
The probability of drawing a club is 13/52 since there are 13 clubs in a standard deck.
The probability of drawing a four is 4/52.
The overall probability is (13/52) * (4/52) = 1/52
b. Without replacement:
The probability of drawing a club in the first draw is 13/52.
After drawing a club, the probability of drawing a four is 4/51 since there are only four fours left in the remaining 51 cards.
The overall probability is (13/52) * (4/51) = 1/51
Now, comparing the calculated probabilities to the given answers, we can see that the answers are correct:
1. (a) 1/4 is correct for with replacement, and (b) 13/51 is correct for without replacement.
2. (a) 1/676 is correct for with replacement, and (b) 1/663 is correct for without replacement.
3. (a) 3/169 is correct for with replacement, and (b) 4/221 is correct for without replacement.
4. (a) 1/52 is correct for with replacement, and (b) 1/51 is correct for without replacement.