If you draw two cards from a standard deck of 52 cards without replacement, find:
P(King first, Jack second)
To find the probability of drawing a King first and a Jack second from a standard deck of 52 cards without replacement, we need to determine two things:
1. The number of ways to choose a King as the first card.
2. The number of ways to choose a Jack as the second card, after drawing a King first.
Step 1: Determine the number of ways to choose a King as the first card.
There are 4 Kings in a standard deck of 52 cards. Therefore, there are 4 ways to choose a King as the first card.
Step 2: Determine the number of ways to choose a Jack as the second card, after drawing a King first.
After drawing a King as the first card, we are left with 51 cards in the deck. Among these 51 cards, there are 4 Jacks. Therefore, there are 4 ways to choose a Jack as the second card.
Step 3: Calculate the probability.
The total number of possible two-card combinations from a 52-card deck is given by the "choose" function, denoted as C(n, r), which calculates the number of combinations of selecting r items from a set of n items. In this case, n = 52 and r = 2, so C(52, 2) = 52! / (2! * (52 - 2)!) = 52! / (2! * 50!).
Now we can calculate the probability:
P(King first, Jack second) = (number of ways to choose a King first) * (number of ways to choose a Jack second) /
(total number of possible two-card combinations)
P(King first, Jack second) = (4 * 4) / C(52, 2) = 16 / (52! / (2! * 50!))
Simplifying the expression may be challenging due to the large numbers involved.
To find the probability of drawing a King first and then a Jack, without replacement from a standard deck of 52 cards, we need to consider the total number of possible outcomes, as well as the number of favorable outcomes.
Step 1: Determine the total number of possible outcomes:
When drawing a card from a deck without replacement, the total number of possible outcomes decreases by 1 with each draw. In this case, the first card can be any of the 52 cards in the deck, and the second card can be any of the remaining 51 cards. Therefore, the total number of possible outcomes is 52 * 51.
Step 2: Determine the number of favorable outcomes:
Since we want to draw a King first and then a Jack, there is only one King among the 52 cards, and once the King is drawn, there are four Jacks remaining in the deck. So, the number of favorable outcomes is 1 * 4 = 4.
Step 3: Calculate the probability:
The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes:
P(King first, Jack second) = Number of favorable outcomes / Total number of possible outcomes
P(King first, Jack second) = 4 / (52 * 51)
P(King first, Jack second) ≈ 0.006
Therefore, the probability of drawing a King first and then a Jack is approximately 0.006 or 0.6%.