Using a standar deck of 52 cards, what is the probability of drawing four cards of the same number(eg.four aces, 4 kings, four 2s, four 3s,etc?
To find the probability of drawing four cards of the same number from a standard deck of 52 cards, we need to determine two things: the number of favorable outcomes and the total number of possible outcomes.
1. Determine the number of favorable outcomes:
Since we need to draw four cards of the same number, we will consider each number from 2 to Ace individually. Each number has four cards in the deck (one of each suit). So, the number of favorable outcomes for each number is 1.
2. Determine the total number of possible outcomes:
To calculate the total number of possible outcomes, we need to consider that we are drawing four cards from a deck of 52 cards. We can use the combinations formula to calculate this value. The formula for combinations is nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items being chosen.
In this case, n = 52 (total number of cards in a deck) and r = 4 (number of cards being drawn). Plugging in these values, we get nCr = 52! / (4!(52-4)!).
Calculating 52!: 52! = 52 × 51 × 50 × ... × 4 × 3 × 2 × 1 (factorial).
Calculating 4!: 4! = 4 × 3 × 2 × 1 (factorial).
Calculating (52-4)! = 48! = 48 × 47 × 46 × ... × 4 × 3 × 2 × 1 (factorial).
Now, let's calculate the value of nCr using the above values.
nCr = 52! / (4!(52-4)!) = (52 × 51 × 50 × ... × 4 × 3 × 2 × 1) / (4 × 3 × 2 × 1) × (48 × 47 × 46 × ... × 4 × 3 × 2 × 1)
Simplifying the equation, we can cancel out similar terms, leaving us with:
nCr = (52 × 51 × 50 × 49)/(4 × 3 × 2 × 1) = 270,725
3. Calculate the probability:
The probability is given by the number of favorable outcomes divided by the number of total possible outcomes. In this case, the probability is equal to 1 (number of favorable outcomes) divided by 270,725 (total possible outcomes).
Therefore, the probability of drawing four cards of the same number from a standard deck of 52 cards is approximately 0.000369 or 0.0369%.