Suppose that 4 cards are drawn from a well-shuffled deck of 52 cards. What is the probability that all 4 are hearts?
Would the answer be 1/4, if I reduce it?
Thanks
C(13,4)/C(52/4)
= 715/270725
= 11/4165 or appr .002641
Nonetheless' second answer is not correct, should be
(13/52)(12/51)(11/50)(10/49)
which comes out to the same as my answer above
"drawing" cards usually means that you are holding the cards in your hand, thus they would not be returned after a draw.
Thanks for catching the denominator errors.
(Drawing can mean either with or without replacement; depends on the textbook, country, etc. so always good to clarify)
To find the probability that all 4 cards drawn are hearts, you need to calculate the number of favorable outcomes (drawing all hearts) divided by the total number of possible outcomes (drawing any 4 cards from the deck).
First, let's determine the number of favorable outcomes. There are 13 hearts in a standard deck of 52 cards, so the number of ways to choose 4 hearts is given by the combination formula: C(13, 4) = 13! / (4!(13-4)!) = 715.
Next, let's determine the total number of possible outcomes. Since we are drawing 4 cards from a deck of 52, the total number of possible outcomes is given by the combination formula: C(52, 4) = 52! / (4!(52-4)!) = 270,725.
Finally, to find the probability, divide the number of favorable outcomes by the total number of possible outcomes:
Probability = Favorable outcomes / Total outcomes = 715 / 270,725 ≈ 0.0026 or 0.26%.
So, the probability that all 4 cards drawn are hearts is approximately 0.26% or 1/385.
No, reducing it to 1/4 would not be correct. The probability is significantly smaller than 1/4 since there are many more ways to draw 4 cards that are not all hearts.
With replacement:
(13/52)(13/52)(13/52)(13/52)
=(1/4)^4
= ?
Without replacement:
(13/52)(12/52)(11/52)(10/52)
= ?