If you are dealt a 5-card hand from a standard deck of 52 playing cards, what is the
probability (rounded to 6 decimal places) that you are dealt at least 4 hearts?
total of hands = C(52,5) = 2598960
at least 4 hearts ---> 4 hearts or 5 hearts
prob(at least 4 hearts) = ( C(39,1)*C(13,4) + C(39,0)*C(13,5) )/2568960
= .011224489
or
= .011224 correct to 6 decimals
To find the probability of being dealt at least 4 hearts in a 5-card hand from a standard deck, we need to calculate the number of favorable outcomes (hands with at least 4 hearts) divided by the total number of possible outcomes (all possible 5-card hands).
First, let's determine the number of favorable outcomes:
1. Calculate the number of ways to choose exactly 4 hearts from the 13 hearts in the deck:
- This can be done using the combination formula, denoted as C(n, k), which represents the number of ways to choose k items from a set of n items without regard to the order.
- In this case, we have n = 13 hearts and k = 4 hearts.
- Thus, the number of ways to choose 4 hearts is C(13, 4) = 13! / (4! * (13 - 4)!) = 715.
2. Calculate the number of ways to choose the remaining 1 card from the remaining 39 cards that are not hearts:
- There are 39 cards remaining after choosing the 4 hearts.
- Thus, the number of ways to choose 1 card is 39.
3. Therefore, the number of favorable outcomes is 715 * 39 = 27885.
Next, let's determine the total number of possible outcomes:
1. Calculate the number of ways to choose any 5 cards from the 52-card deck:
- This can be done using the combination formula, denoted as C(n, k), where n = 52 cards and k = 5 cards.
- Thus, the number of ways to choose 5 cards is C(52, 5) = 52! / (5! * (52 - 5)!) = 2598960.
Now, we can calculate the probability:
P(at least 4 hearts) = number of favorable outcomes / total number of possible outcomes
= 27885 / 2598960
≈ 0.010714 (rounded to 6 decimal places)
Therefore, the probability of being dealt at least 4 hearts is approximately 0.010714 or 1.0714%.