three cards are drawn from a deckof playing cards.Find the probability that the firstcard is red,the second cardis a black king,and the third card is a black king.
To find the probability, we need to first determine the total number of possible outcomes and then determine the number of favorable outcomes.
Total number of possible outcomes:
When three cards are drawn from a deck of playing cards without replacement, the total number of possible outcomes is determined by the number of ways the three cards can be chosen out of 52 cards. This can be calculated using combinations.
Denoting the combination of choosing r objects out of a set of n objects as C(n, r), the total number of possible outcomes can be calculated as:
Total outcomes = C(52, 3) = 52! / (3!(52-3)!) = 22,100
Number of favorable outcomes:
1. The probability that the first card is red:
There are 26 red cards in a deck of playing cards. After choosing one, there are 51 cards left.
Favorable outcomes = 26
2. The probability that the second card is a black king:
There are two black kings in a deck of playing cards. After choosing one, there are 50 cards left.
Favorable outcomes = 2
3. The probability that the third card is a black king:
After the first two cards are drawn, there is only one black king remaining in the deck. So,
Favorable outcomes = 1
Now, we multiply the number of favorable outcomes together to get the total number of favorable outcomes:
Total favorable outcomes = (26) * (2) * (1) = 52
Finally, we calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Total favorable outcomes / Total outcomes = 52 / 22,100 ≈ 0.00235
Therefore, the probability that the first card is red, the second card is a black king, and the third card is a black king is approximately 0.00235 or 0.235%.