IF YOU ARE DEALT 5 CARDS FROM A SHUFFLED DECK OF 52 CARDS, FIND THE PROBABILITY THAT ALL 5 CARDS ARE HEARTS. FIND THE PROBABILITY THAT 4 CARDS ARE HEARTS.

first card 1/4

second card 12/51
third card 11/50
fourth card 10/49
fifth card 9/48
multiply :)

that comes out to:

.0004952

or use combinations
13 hearts in deck
number of combinations of 5 in 13 =
c(13,5) = 13!/[ 5!(8!) ] = 1,287
5 in 52 = 52!/[5!(47!)] = 2,598,960
quotient = .0004952

To find the probability that all 5 cards are hearts, we need to calculate the number of favorable outcomes (5 hearts) divided by the number of possible outcomes (5 cards drawn from a deck of 52 cards).

Step 1: Calculate the number of favorable outcomes (5 hearts):
There are 13 hearts in a deck of 52 cards, so the number of ways to choose 5 hearts out of 13 is denoted as C(13,5) or "13 choose 5." This can be calculated using the combination formula: C(13,5) = 13! / (5! * (13-5)!).

Step 2: Calculate the number of possible outcomes (5 cards drawn from a deck of 52 cards):
There are 52 cards in a deck, so the number of ways to choose 5 cards out of 52 is denoted as C(52,5) or "52 choose 5." This can be calculated using the combination formula: C(52,5) = 52! / (5! * (52-5)!).

Step 3: Calculate the probability of getting all 5 cards as hearts:
The probability is calculated by dividing the number of favorable outcomes by the number of possible outcomes: P(5 hearts) = C(13,5) / C(52,5).

For the probability that 4 cards are hearts, we can follow a similar process, but with a slight difference.

Step 1: Calculate the number of favorable outcomes (4 hearts):
There are 13 hearts in a deck of 52 cards. We need to choose 4 hearts out of 13, which can be calculated as C(13,4) = 13! / (4! * (13-4)!).

Step 2: Calculate the number of possible outcomes (5 cards drawn from a deck of 52 cards) - the same as before: C(52,5) = 52! / (5! * (52-5)!).

Step 3: Calculate the probability of getting 4 cards as hearts:
The probability is calculated by dividing the number of favorable outcomes by the number of possible outcomes: P(4 hearts) = C(13,4) / C(52,5).

Now you can use these formulas to find the probabilities.

To find the probability of all 5 cards being hearts, we need to calculate the ratio of favorable outcomes to the total number of possible outcomes.

There are 13 hearts in a deck of 52 cards, so the number of ways to choose 5 hearts from the deck is given by the combination formula:

C(13, 5) = 13! / (5!(13-5)!) = 1287

The total number of ways to choose 5 cards from a deck of 52 is given by:

C(52, 5) = 52! / (5!(52-5)!) = 259,896

So, the probability of all 5 cards being hearts is:

Probability = favorable outcomes / total outcomes = 1287 / 259,896 ≈ 0.00496 or 0.496%

Now, let's find the probability of 4 cards being hearts:

There are 13 hearts, and we want to choose 4 of them. The remaining card could be any card other than a heart, so we have 39 choices.

The number of ways to choose 4 hearts from 13 is given by:

C(13, 4) = 13! / (4!(13-4)!) = 715

The number of ways to choose the remaining card from the 39 non-heart cards is given by:

C(39, 1) = 39! / (1!(39-1)!) = 39

So, the total number of favorable outcomes is:

Favorable outcomes = C(13, 4) * C(39, 1) = 715 * 39 = 27,885

Therefore, the probability of 4 cards being hearts is:

Probability = favorable outcomes / total outcomes = 27,885 / 259,896 ≈ 0.10748 or 10.748%