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Chapter 2: Problem 119

If 4 cards are drawn at random and without replacement from a deck of 52 playing cards, what is the chance of drawing the 4 aces as the first 4 cards?

Short Answer

Expert verified
The chance of drawing the 4 aces as the first 4 cards is approximately 0.000369% or 1 in 270,725.

Step by step solution

01

Determine the number of successful outcomes

In this case, there is only one successful outcome: we draw all 4 aces in the first 4 draws. Thus, the number of successful outcomes is 1.
02

Determine the total number of possible outcomes

We can compute the total possible outcomes by finding the number of ways we can draw 4 cards from a deck of 52. This can be calculated using combinations, since the order of cards does not matter: \( \binom {52}{4} = \frac{52!}{4!(52-4)!} \) Where: 52! is the factorial of 52 (52*51*50*...*1) 4! is the factorial of 4 (4*3*2*1) (52-4)! is the factorial of (52-4=48)
03

Compute the combinations

Now, using the given formula we can compute the combinations: \( \binom {52}{4} = \frac{52!}{4!(48)!} = \frac{52\times51\times50\times49}{4\times3\times2\times1} = 270,725 \)
04

Calculate the probability

Now that we have the number of successful outcomes (1) and the total number of possible outcomes (270,725), we can calculate the probability using the formula: \( P(A) = \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}} \) \( P(\text{Drawing all 4 aces in the first 4 draws}) = \frac{1}{270,725} \)
05

Simplify and express the probability as a percentage, fraction, or decimal

The probability can be expressed in different forms: As a fraction: \( \frac{1}{270,725} \) As a decimal: \( \approx 0.00000369 \) As a percentage: \( \approx 0.000369 \% \) So, the chance of drawing the 4 aces as the first 4 cards is approximately 0.000369% or 1 in 270,725.

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