Five cards are dealt from a well-shuffled standard deck. using combinations, write an expresssion for the probability all are hearts
To find the probability of all five cards being hearts, we need to calculate the number of favorable outcomes and the total number of possible outcomes.
First, let's determine the total number of possible outcomes when dealing five cards from a standard deck. For each card, we have 52 options available, so the total number of possible outcomes is 52 × 51 × 50 × 49 × 48 = 311,875,200.
Next, let's calculate the number of favorable outcomes. We want all five cards to be hearts, so we can choose 5 hearts out of the 13 available in the deck, which can be done using combinations. The number of ways to choose 5 hearts from 13 is denoted by the notation "C(13, 5)" and can be calculated as:
C(13, 5) = 13! / (5! * (13 - 5)!) = 13! / (5! * 8!) = (13 × 12 × 11 × 10 × 9) / (5 × 4 × 3 × 2 × 1) = 1287.
Finally, we divide the number of favorable outcomes by the total number of possible outcomes to get the probability:
P(all cards are hearts) = number of favorable outcomes / total number of outcomes
= 1287 / 311,875,200
Simplifying further may not be necessary, as the resulting probability is approximately:
P(all cards are hearts) ≈ 4.13 × 10^(-6)
To find the probability of drawing all hearts when five cards are dealt from a well-shuffled standard deck, we need to determine the number of favorable outcomes and the total number of possible outcomes.
The total number of possible outcomes can be calculated using combinations. There are 52 cards in a deck, and we need to choose 5 cards, so the total number of possible outcomes is given by the combination formula:
C(52, 5) = 52! / (5! * (52 - 5)!)
To calculate the number of favorable outcomes, we need to consider that there are 13 hearts in a deck. We need to choose all 5 cards from the 13 hearts, so the number of favorable outcomes is given by the combination formula:
C(13, 5) = 13! / (5! * (13 - 5)!)
Therefore, the probability of drawing all hearts when five cards are dealt can be expressed as:
C(13, 5) / C(52, 5) = (13! / (5! * (13 - 5)!)) / (52! / (5! * (52 - 5)!))
Simplifying this expression will give you the numerical value of the probability.