Probability of Drawing Two Jacks from a Standard Deck of 52 Cards

In probability theory, we often study the likelihood of certain events occurring. One such classic problem is calculating the probability of drawing two Jacks when three cards are drawn randomly from a standard deck of 52 cards. This article will guide you through the step-by-step process of determining this probability using combinatorial methods. By the end, you will have a thorough understanding of the underlying mathematics and be able to apply similar techniques to other probabilistic scenarios.

Step-by-Step Calculation

Step 1: Determine the Total Number of Ways to Draw 3 Cards

The total number of ways to draw 3 cards from 52 can be calculated using the combination formula:

Combination Formula

( binom{n}{k} frac{n!}{k!(n-k)!} )

Here, ( n 52 ) and ( k 3 ). Plugging these values into the formula, we get:

Calculation

( binom{52}{3} frac{52!}{3!(52-3)!} frac{52 times 51 times 50}{3 times 2 times 1} 22100 )

Step 2: Determine the Number of Favorable Outcomes

For exactly two Jacks in the three drawn cards, we need to choose:

2 Jacks from the 4 available Jacks: ( binom{4}{2} ) 1 non-Jack card from the remaining 48 cards: ( binom{48}{1} )

Computation of Favorable Outcomes

Choosing 2 Jacks from 4:

( binom{4}{2} frac{4!}{2!(4-2)!} frac{4 times 3}{2 times 1} 6 )

Choosing 1 non-Jack card from 48 cards:

( binom{48}{1} 48 )

Now, multiply the two results:

Favorable outcomes ( 6 times 48 288 )

Step 3: Calculate the Probability

The probability of drawing exactly 2 Jacks in 3 cards is the ratio of favorable outcomes to the total outcomes:

( P frac{288}{22100} )

Step 4: Simplifying the Probability

Let us simplify ( frac{288}{22100} ).

Find the greatest common divisor (GCD) of 288 and 22100:

The GCD is 4.

Hence, the simplified probability is:

( P frac{288 div 4}{22100 div 4} frac{72}{5525} )

Final Result

Therefore, the probability of drawing exactly 2 Jacks when 3 cards are drawn from a deck of 52 cards is:

( boxed{frac{72}{5525}} approx 0.0130 ) or 1.30%

Using R for Hypergeometric Distribution

In statistics, the hypergeometric distribution can be used to model scenarios like drawing cards from a deck. The hypergeometric distribution formula can be applied in R to confirm our calculation:

R Code

library(stats)
The output returns the probability:
[
1] 0.0001809955
]