Three cards are drawn from a deck without replacement. Find these probabilities. a)all are jacks b)all are clubs c)all are red cards
However, each of the above has to occur three times. Remember that the cards are not replaced.
a) 4/52 * 3/51 * 2/50 = ?
Use the same principle for the other two problems.
a) Well, if you want all the jacks, you might have better luck fishing in a box of cereal. But let's calculate the probability anyway. Since there are 4 jacks in a deck of 52 cards, the probability of drawing the first jack is 4/52. After that, there are 3 jacks left out of 51 cards, so the probability of drawing the second jack is 3/51. Finally, there are 2 jacks left out of the remaining 50 cards, so the probability of drawing the third jack is 2/50. Multiplying all these probabilities together, we get (4/52) * (3/51) * (2/50) = 1/5525. So, the probability of drawing all jacks is about as likely as spotting a unicorn at the mall.
b) If you want all clubs, I hope you have a good sense of humor because the probability is going to be a joke! There are 13 clubs in a deck of 52 cards, so the probability of drawing the first club is 13/52. Then, there are only 12 clubs left out of 51 cards, so the probability of drawing the second club is 12/51. Finally, there are just 11 clubs left out of the remaining 50 cards, so the probability of drawing the third club is 11/50. Multiplying all these probabilities together, we get (13/52) * (12/51) * (11/50) = 0.0112. So, the probability of drawing all clubs is about as likely as finding a penguin in the Sahara desert.
c) Now, all red cards, huh? Well, let's give it a whirl. There are 26 red cards in a deck of 52 cards, so the probability of drawing the first red card is 26/52. After that, there are 25 red cards left out of 51 cards, so the probability of drawing the second red card is 25/51. Finally, there are 24 red cards left out of the remaining 50 cards, so the probability of drawing the third red card is 24/50. Multiplying all these probabilities together, we get (26/52) * (25/51) * (24/50) = 0.1176. So, the probability of drawing all red cards is about as likely as finding a four-leaf clover in a field of dandelions.
To find the probabilities, we need to consider the number of favorable outcomes (desired outcomes) and the number of possible outcomes (total outcomes).
a) Probability of drawing all jacks:
There are 4 jacks in the deck of 52 cards.
So, the number of favorable outcomes is C(4,3) = 4.
The total number of outcomes is C(52,3) = 22,100.
Therefore, the probability is 4/22,100.
b) Probability of drawing all clubs:
There are 13 clubs in the deck of 52 cards.
So, the number of favorable outcomes is C(13,3) = 286.
The total number of outcomes is C(52,3) = 22,100.
Therefore, the probability is 286/22,100.
c) Probability of drawing all red cards:
There are 26 red cards in the deck of 52 cards (13 hearts and 13 diamonds).
So, the number of favorable outcomes is C(26,3) = 2,600.
The total number of outcomes is C(52,3) = 22,100.
Therefore, the probability is 2,600/22,100.
Hence, the probabilities are:
a) 4/22,100
b) 286/22,100
c) 2,600/22,100
To find the probabilities, we need to know the total number of possible outcomes and the number of favorable outcomes for each case.
a) All three cards are jacks:
There are 4 jacks in a deck of 52 cards. When we draw one jack, the number of jacks decreases to 3 out of 51 cards. Similarly, after drawing a second jack, there would be 2 out of 50 jacks remaining, and after the third draw, there would be 1 out of 49 jacks remaining.
The probability of drawing all jacks can be calculated as:
P(all jacks) = (4/52) * (3/51) * (2/50) * (1/49)
b) All three cards are clubs:
There are 13 clubs in a deck of 52 cards. Similar to the previous case, the probability for each draw changes after each card is drawn.
P(all clubs) = (13/52) * (12/51) * (11/50) * (10/49)
c) All three cards are red cards:
There are 26 red cards (13 hearts + 13 diamonds) in a deck of 52 cards.
P(all red cards) = (26/52) * (25/51) * (24/50) * (23/49)
Now, by performing the necessary calculations, we can find the desired probabilities.
52 cards in a deck.
4/52 Jacks = 1/13
13/52 Clubs = 1/4
26/52 Red = 1/2