Assume that you randomly select 8 cards from a deck of 52.
What is the probability that all of the cards selected are hearts?
Do you replace each card after your select it? If so, P = (13/52)^8 = ?
If not, P = 13/52 * 12/51 * 11/50 * 10/49 * 9/48 * 8/47 * 7/46 * 6/45 = ?
Do you think you could find a rule for the second probability?
aaah waet
To find the probability that all of the cards selected are hearts, we need to determine two things: the number of favorable outcomes (i.e., the number of ways to select all hearts) and the total number of possible outcomes (i.e., the total number of ways to select any 8 cards from the deck).
Step 1: Calculate the number of favorable outcomes:
There are 13 hearts in a deck of 52 cards. We want to select all 8 hearts, so the number of favorable outcomes is given by the combination formula:
C(13, 8) = 13! / (8! * (13-8)!) = 1287
Step 2: Calculate the total number of possible outcomes:
The total number of ways to select any 8 cards from a deck of 52 is given by the combination formula as well:
C(52, 8) = 52! / (8! * (52-8)!) = 752,538,150
Step 3: Calculate the probability:
The probability is given by the ratio of the number of favorable outcomes to the total number of possible outcomes:
P(all cards selected are hearts) = favorable outcomes / total outcomes = 1287 / 752,538,150 ≈ 0.00000171
Therefore, the probability that all of the cards selected are hearts is approximately 0.00000171, or 0.000171%.
To find the probability of randomly selecting all hearts, we need to determine the total number of favorable outcomes (selecting 8 hearts) and divide it by the total number of possible outcomes (selecting any 8 cards from the deck).
1. Determine the total number of favorable outcomes: In a standard deck of 52 cards, there are 13 hearts. Since we want to select all 8 hearts, we need to calculate the number of ways to choose 8 cards out of the 13 hearts. This can be done using the combination formula, often denoted as "nCr", which calculates the number of ways to choose r objects from a total of n objects without regard to the order.
In this case, we want to choose 8 hearts from a total of 13, so we calculate the combination of 13 hearts taken 8 at a time:
nCr = n! / (r! * (n - r)!)
C(13, 8) = 13! / (8! * (13 - 8)!)
Simplifying the formula:
C(13, 8) = 13! / (8! * 5!)
Now we can calculate the combination:
C(13, 8) = (13 * 12 * 11 * 10 * 9) / (5 * 4 * 3 * 2 * 1)
C(13, 8) = 1287
Therefore, there are 1287 ways to select 8 hearts from a deck of 13 hearts.
2. Determine the total number of possible outcomes: To find the total number of ways to select any 8 cards from the 52-card deck, we can calculate the combination of 52 cards taken 8 at a time:
C(52, 8) = 52! / (8! * (52 - 8)!)
Simplifying the formula:
C(52, 8) = 52! / (8! * 44!)
Now we can calculate the combination:
C(52, 8) = (52 * 51 * 50 * 49 * 48 * 47 * 46 * 45) / (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
C(52, 8) = 752538150
Therefore, there are 752538150 ways to select any 8 cards from a deck of 52.
3. Calculate the probability: The probability of selecting all 8 hearts is the ratio of favorable outcomes to total outcomes:
Probability = favorable outcomes / total outcomes
Probability = 1287 / 752538150
Simplifying the fraction, we get:
Probability ≈ 0.00000171
Therefore, the probability of randomly selecting all hearts from a deck of 52 cards is approximately 0.00000171, or about 1 in 585,009.