Two cards are selected from a standard deck of 52 playing cards. The first card is not replaced before the second card is selected. Find the probability of selecting a diamond and then selecting a spade. the probability of selecting a diamond and then selecting a spade is?
Well, let's see. A diamond and a spade walk into a bar...
But seriously, to find the probability, we need to determine the number of favorable outcomes and divide it by the total number of possible outcomes.
The number of favorable outcomes is calculated by considering that there are 13 diamonds in a deck of 52 cards, and after selecting one, there are 12 diamonds left. Similarly, there are 13 spades in the deck, and after selecting one, there are 12 spades left. So, the number of favorable outcomes is 13 multiplied by 12, which is 156.
Now, the total number of possible outcomes is the number of ways to choose any two cards from a deck of 52 cards, which is calculated by using combinations. It's written as 52 choose 2 and can be calculated as 52! ÷ (2!(52-2)!), which simplifies to 52! ÷ (2!50!).
Using some mathematical magic, we find that the total number of possible outcomes is 1,326.
Now, to find the probability, we divide the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of Favorable Outcomes ÷ Total Number of Possible Outcomes
= 156 ÷ 1,326
≈ 0.1176 or 11.76%
So, the probability of selecting a diamond and then selecting a spade is approximately 11.76%.
To find the probability of selecting a diamond and then selecting a spade, we can break it down into two steps:
Step 1: Probability of selecting a diamond on the first draw.
Step 2: Probability of selecting a spade on the second draw after the diamond is removed.
Step 1:
In a standard deck of 52 playing cards, there are 13 diamonds. So, the probability of drawing a diamond on the first draw is:
P(diamond on first draw) = Number of favorable outcomes / Total number of outcomes
= 13 diamonds / 52 cards
= 1/4
Step 2:
Since the first card is not replaced, after removing the diamond from the deck, there are now 51 cards left, including 12 diamonds and 13 spades.
Therefore, the probability of selecting a spade on the second draw after removing the diamond is:
P(spade on second draw) = Number of favorable outcomes / Total number of outcomes
= 13 spades / 51 cards
Now, we can multiply the probabilities of both steps to find the overall probability:
P(diamond and then spade) = P(diamond on first draw) x P(spade on second draw)
= (1/4) x (13/51)
= 13/204
Thus, the probability of selecting a diamond and then selecting a spade is 13/204.
To find the probability of selecting a diamond and then selecting a spade, we can use the concept of conditional probability.
Step 1: Determine the number of favorable outcomes:
There are 13 diamonds in a standard deck of 52 playing cards, so the probability of selecting a diamond on the first draw is 13/52.
After selecting a diamond (assuming it was not replaced), there are now 51 cards remaining in the deck, with 13 of them being spades. Therefore, the probability of selecting a spade on the second draw, given that a diamond was selected first, is 13/51.
Step 2: Calculate the probability:
To find the overall probability, we multiply the probabilities of the individual events:
P(diamond and then spade) = (13/52) * (13/51)
Simplifying this expression:
P(diamond and then spade) = (1/4) * (13/51)
P(diamond and then spade) = 13/204
Hence, the probability of selecting a diamond and then selecting a spade is 13/204.