Assume that you randomly select 8 cards from a deck of 52.
What is the probability that all of the cards selected are hearts?
56
To find the probability of drawing all hearts when selecting 8 cards randomly from a standard deck of 52 cards, we first need to determine the number of favorable outcomes and the total number of possible outcomes.
In a standard deck, there are 13 hearts (since there are 13 cards of each suit). So, the number of favorable outcomes is the number of ways you can choose all 8 hearts from the 13 hearts available.
To calculate this, we can use the combination formula:
nCr = n! / (r!(n-r)!)
In this case, we want to choose 8 hearts from a pool of 13 hearts.
So, the number of favorable outcomes is 13C8 = 13! / (8!(13-8)!), which simplifies to 13! / (8!5!).
Now, let's calculate it step by step:
13! = 13 * 12 * 11 * 10 * 9 * 8!
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
Simplifying the factorials, we get:
13! = 13 * 12 * 11 * 10 * 9 * 8!
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
These common terms cancel out, leaving us with:
13C8 = (13 * 12 * 11 * 10 * 9) / (5 * 4 * 3 * 2 * 1)
Calculating this expression, we get:
13C8 = 1287
Now, let's determine the total number of possible outcomes, which is the number of ways we can choose any 8 cards from the 52 cards in the deck.
Using the same combination formula, we have:
52C8 = 52! / (8!(52-8)!), which simplifies to 52! / (8!44!).
52! = 52 * 51 * 50 * 49 * 48 * 47 * 46 * 45 * 44!
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
44! = 44 * 43 * 42 * 41 * 40 * 39 * 38 * 37 * 36 * 35 * 34 * 33 * 32 * 31 * 30 * 29 * 28 * 27 * 26 * 25 * 24 * 23 * 22 * 21 * 20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
Calculating these expressions, we get:
52! = a very large number
8! = 40320
44! = a very large number
52C8 = (52 * 51 * 50 * 49 * 48 * 47 * 46 * 45) / 40320
Simplifying, we get:
52C8 = 752538150
Finally, to calculate the probability, we divide the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 1287 / 752538150
To obtain the decimal value, we divide 1287 by 752538150 to get approximately 0.00000171.
So, the probability of drawing all hearts when selecting 8 cards randomly from a standard deck of 52 cards is approximately 0.00000171.