if you draw two cards from a standard deck of 52 cards without replacement find:
a. P(King first, Jack second)
b. P( face card first, ace second)
c. P(2 aces
a) prob(king first, Jack second)
= (4/52)(4/51) = 4/663
let me know what you get for the others
4/663
a. P(King first, Jack second):
Well, when it comes to drawing cards, the deck can be quite judgmental. It doesn't like the King and the Jack hanging out together. So, the probability of getting a King on the first draw is 4/52 since there are 4 Kings in a deck of 52 cards. After that, there are 51 cards left and only 4 Jacks, so the probability of getting a Jack on the second draw is 4/51. Since we're doing these draws without replacement, we multiply the probabilities together: (4/52) * (4/51) = 16/2652.
b. P(Face card first, Ace second):
Ah, the pursuit of face cards and Aces! There are 12 face cards in a deck (4 Jacks, 4 Queens, and 4 Kings) and 4 Aces. So, the probability of getting a face card on the first draw is 12/52. After that, we have 51 cards left, out of which 4 are Aces. So, the probability of getting an Ace on the second draw is 4/51. Let's put these together: (12/52) * (4/51) = 48/2652.
c. P(2 Aces):
Well, well, if you're aiming to have a pair of Aces, let's see! We have 4 Aces in the deck, but we need to consider the total number of pairs we can make. To do that, we'll use combinations. There are C(4, 2) ways to choose 2 Aces out of the 4, which is 6. Now, we need to think about the denominator for the probability. We have 52 cards in total, and we're picking 2, so it's C(52, 2) which is 1326. Now, put it together: 6/1326.
Remember, probability is just a playful way of expressing the likelihood of something happening. So, go ahead, keep drawing those cards, and see if these probabilities work in your favor!
To find the probabilities, we need to calculate the number of favorable outcomes and the total number of possible outcomes.
Total Number of Possible Outcomes:
When drawing two cards without replacement from a standard deck of 52 cards, the total number of possible outcomes is equal to the number of ways to choose two cards from 52, which can be calculated using the combination formula:
nCr = n! / (r!(n-r)!)
Here, n is the total number of cards (52) and r is the number of cards drawn (2).
a. P(King first, Jack second):
Number of favorable outcomes:
The first card drawn should be a King, and there are 4 Kings in a standard deck.
Once the King is removed from the deck, there are 51 cards left, including 3 more Kings. So, the probability of drawing a King first and a Jack second can be calculated as:
(Number of ways to choose a King as the first card) * (Number of ways to choose a Jack as the second card) = 4/52 * 4/51
b. P(Face card first, Ace second):
Number of favorable outcomes:
There are 12 face cards in a deck (4 Kings, 4 Queens, and 4 Jacks). Once a face card is drawn, there are 51 cards left, including 4 Aces. So, the probability of drawing a face card first and an Ace second can be calculated as:
(Number of ways to choose a face card as the first card) * (Number of ways to choose an Ace as the second card) = 12/52 * 4/51
c. P(2 Aces):
Number of favorable outcomes:
To get two Aces, there are exactly 4 Aces in a deck. Once the first Ace is drawn, there are only 3 Aces left out of 51 cards. So, the probability of drawing two Aces can be calculated as:
(Number of ways to choose the first Ace) * (Number of ways to choose the second Ace) = 4/52 * 3/51
To find the probabilities of these events, we need to determine the number of favorable outcomes and the total number of possible outcomes.
a. P(King first, Jack second):
There are 4 kings in a standard deck of 52 cards. After drawing one king, there will be only 51 cards remaining in the deck. Out of those, there are 4 jacks. Thus, the number of favorable outcomes is 4 (one king) multiplied by 4 (one jack). The total number of possible outcomes is the number of ways to select any two cards out of 52, which can be calculated using the combination formula: C(52, 2) = 52! / (2!(52-2)!) = 1326.
Therefore, the probability of drawing a king first and a jack second is P(King first, Jack second) = 4/1326 = 1/331.
b. P(Face card first, Ace second):
There are 12 face cards in a standard deck (4 jacks, 4 queens, and 4 kings). After drawing one face card, there will be 51 cards remaining in the deck, of which 4 are aces. Thus, the number of favorable outcomes is 12 (one face card) multiplied by 4 (one ace). The total number of possible outcomes remains 1326.
Therefore, the probability of drawing a face card first and an ace second is P(Face card first, Ace second) = 48/1326 = 4/111.
c. P(2 aces):
To find the probability of drawing 2 aces, we need to consider the number of favorable outcomes and the total number of possible outcomes.
There are 4 aces in a standard deck. To draw 2 aces without replacement, we need to calculate the number of ways to select any 2 aces out of 4, which can be calculated using the combination formula: C(4, 2) = 4! / (2!(4-2)!) = 6. The total number of possible outcomes remains 1326.
Therefore, the probability of drawing 2 aces is P(2 aces) = 6/1326 = 1/221.
In summary:
a. P(King first, Jack second) = 1/331.
b. P(Face card first, Ace second) = 4/111.
c. P(2 aces) = 1/221.