what is the probability that a hand of five cards contains only heart?
The probability that a hand of five cards contains only hearts can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
Total number of possible outcomes:
There are 52 cards in a standard deck. So, the total number of possible outcomes is the number of ways to choose 5 cards from a deck of 52. This can be calculated using combinations and expressed as:
52 C 5 = 52! / (5! * (52-5)!) = 2,598,960
Number of favorable outcomes:
There are 13 hearts in a standard deck, so we need to choose 5 cards out of these 13 hearts. Using combinations, this can be expressed as:
13 C 5 = 13! / (5! * (13-5)!) = 1,287
Therefore, the number of favorable outcomes is 1,287.
Now, we can calculate the probability:
Probability = Favorable Outcomes / Total Outcomes
= 1,287 / 2,598,960
≈ 0.000495 or 0.0495%
So, the probability that a hand of five cards contains only hearts is approximately 0.0495% or 1 in 2,020.
To calculate the probability of getting a hand of five cards that contains only hearts, we need to consider two factors: the total number of possible hands and the number of favorable outcomes.
Step 1: Determine the total number of possible hands
In a standard deck of 52 playing cards, there are 13 hearts (one for each rank). So, the total number of possible hands that can be drawn from a deck of cards is given by:
C(52, 5) = 52! / (5!(52 - 5)!) = 2,598,960
Step 2: Determine the number of favorable outcomes
To calculate the number of favorable outcomes, we need to consider that we want all five cards in the hand to be hearts. Since there are 13 hearts in the deck, the number of ways to choose 5 hearts from the 13 is given by:
C(13, 5) = 13! / (5!(13 - 5)!) = 1287
Step 3: Calculate the probability
Now, we divide the number of favorable outcomes by the total number of possible outcomes to determine the probability:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 1287 / 2,598,960
Therefore, the probability of getting a hand of five cards that contains only hearts is approximately 0.000495 or 0.0495%.
To find the probability that a hand of five cards contains only hearts, we need to consider the total number of possible hands and the number of hands that contain only hearts.
Step 1: Calculate the total number of possible hands:
There are 52 cards in a standard deck, and we want to choose 5 cards without replacement. So, the number of possible hands is given by the combination formula:
total number of hands = C(52, 5) = 52! / (5!(52-5)!) = 2,598,960.
Step 2: Calculate the number of hands that contain only hearts:
There are 13 hearts in a deck, so we want to choose 5 hearts without replacement. Using the combination formula again, we can calculate the number of hands that contain only hearts:
number of hands with only hearts = C(13, 5) = 13! / (5!(13-5)!) = 1,287.
Step 3: Calculate the probability:
To find the probability that a hand of five cards contains only hearts, divide the number of hands with only hearts by the total number of hands:
probability = (number of hands with only hearts) / (total number of hands) = 1,287 / 2,598,960 ≈ 0.000495.
Therefore, the probability that a hand of five cards contains only hearts is approximately 0.000495, or 0.0495%.