If you are dealt 3 cards from a shuffled deck of 52 cards, find the probability that all 3 cards are picture cards.
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To find the probability that all 3 cards are picture cards, we need to determine the number of ways we can select 3 picture cards out of the 52 cards in the deck.
Step 1: Determine the number of picture cards in a deck
A standard deck of playing cards contains 12 picture cards, which consist of 3 face cards (Jack, Queen, and King) in each of the 4 suits (hearts, diamonds, clubs, and spades).
Step 2: Determine the number of ways to select 3 picture cards out of 12
Since we want to select 3 picture cards, we can use the combination formula to calculate the number of ways to select 3 out of 12. The formula for combination is:
nCr = n! / (r! * (n-r)!)
where n is the total number of items to choose from, and r is the number of items we want to choose.
Using this formula, we have:
12C3 = 12! / (3! * (12-3)!)
= 12! / (3! * 9!)
= (12 * 11 * 10) / (3 * 2 * 1)
= 220
So, there are 220 ways to select 3 picture cards out of 12.
Step 3: Determine the total number of ways to select 3 cards from a deck of 52 cards
Since we want to select 3 cards from a deck of 52 cards, we can again use the combination formula:
52C3 = 52! / (3! * (52-3)!)
= 52! / (3! * 49!)
= (52 * 51 * 50) / (3 * 2 * 1)
= 22,100
So, there are 22,100 ways to select 3 cards from a deck of 52 cards.
Step 4: Calculate the probability
The probability of selecting 3 picture cards is given by:
Probability = Number of favorable outcomes / Total possible outcomes
Number of favorable outcomes = 220 (from Step 2)
Total possible outcomes = 22,100 (from Step 3)
Probability = 220 / 22,100
= 0.0099502
Therefore, the probability that all 3 cards are picture cards is approximately 0.00995, or about 0.995%.
To find the probability that all three cards are picture cards when dealt from a shuffled deck of 52 cards, you first need to determine the total number of possible outcomes and the number of favorable outcomes.
Step 1: Find the total number of possible outcomes.
Since you are dealt 3 cards from a deck of 52 cards, the total number of possible outcomes can be calculated using the concept of combinations. The number of combinations of 52 cards taken 3 at a time can be denoted as C(52, 3) and can be calculated using the formula:
C(n, k) = n! / (k!(n-k)!)
In this case, the total number of possible outcomes is C(52, 3).
Step 2: Find the number of favorable outcomes.
To find the number of favorable outcomes, you need to determine the number of ways you can select 3 picture cards from the deck. Picture cards include kings, queens, and jacks, and there are 4 suits for each picture card. So, the number of ways to select 3 picture cards can be calculated as:
4C3 (selecting 3 suits) * 4C1 (selecting 1 card from each suit) = C(4,3) * C(4,1)
Step 3: Calculate the probability.
Finally, to calculate the probability, divide the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
Using the values calculated in Steps 1 and 2, you can now solve for the probability.