What is the probability that a randomly drawn 6-cards hand from a standard deck....
B) contains at least four clubs?
C) contains at most three clubs?
1/4 of all cards are clubs. use probability.
To determine the probabilities of obtaining a hand with specific club cards from a standard deck of 52 playing cards, we need to calculate the number of favorable outcomes (hands meeting the specified criteria) divided by the total number of possible outcomes (all possible 6-card hands).
Let's start with part B:
1. Determine the number of favorable outcomes (hands with at least four clubs):
- There are 13 clubs in a standard deck, so the possible combinations of selecting four, five, or six clubs from the 13 available are:
* Four clubs: Select 4 clubs from 13: C(13, 4)
* Five clubs: Select 5 clubs from 13: C(13, 5)
* Six clubs: Select all 6 clubs from 13: C(13, 6)
- To calculate the total number of favorable outcomes, add up the three cases: C(13, 4) + C(13, 5) + C(13, 6).
2. Determine the total number of possible outcomes (all possible 6-card hands):
- There are 52 cards in a deck, and we need to select 6 cards.
- Calculate the total number of possible outcomes: C(52, 6).
3. Calculate the probability:
- Divide the number of favorable outcomes (step 1) by the total number of possible outcomes (step 2).
- Probability = (C(13, 4) + C(13, 5) + C(13, 6)) / C(52, 6).
Now let's move on to part C:
1. Determine the number of favorable outcomes (hands with at most three clubs):
- The possible combinations include cases with 0, 1, 2, or 3 clubs.
* Zero clubs: Select all 6 cards from the non-clubs (39) in the deck: C(39, 6)
* One club: Select 1 club from 13 and 5 non-club cards: C(13, 1) * C(39, 5)
* Two clubs: Select 2 clubs from 13 and 4 non-club cards: C(13, 2) * C(39, 4)
* Three clubs: Select 3 clubs from 13 and 3 non-club cards: C(13, 3) * C(39, 3)
- To calculate the total number of favorable outcomes, add up the four cases: C(39, 6) + C(13, 1) * C(39, 5) + C(13, 2) * C(39, 4) + C(13, 3) * C(39, 3).
2. Determine the total number of possible outcomes (all possible 6-card hands), which remains the same as in step 2 of part B: C(52, 6).
3. Calculate the probability:
- Divide the number of favorable outcomes (step 1) by the total number of possible outcomes (step 2).
- Probability = (C(39, 6) + C(13, 1) * C(39, 5) + C(13, 2) * C(39, 4) + C(13, 3) * C(39, 3)) / C(52, 6).
By following these steps, you can calculate the probabilities for both parts B and C based on the specific criteria.