Six cards are drawn at random from a pack of cards. What is the probability that 3 will be red and 3 black?

Understand the Problem

The question asks to calculate the probability of drawing 3 red cards and 3 black cards when 6 cards are randomly selected from a standard deck of cards.

Answer

$\frac{13000}{39151}$

Steps to Solve

Calculate the total number of ways to choose 6 cards from 52

The total number of ways to choose 6 cards from a standard deck of 52 cards is given by the combination formula:

$$ {52 \choose 6} = \frac{52!}{6!(52-6)!} = \frac{52!}{6!46!} = \frac{52 \times 51 \times 50 \times 49 \times 48 \times 47}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 20,358,520 $$
Calculate the number of ways to choose 3 red cards from 26

There are 26 red cards in a standard deck. The number of ways to choose 3 red cards from 26 is:

$$ {26 \choose 3} = \frac{26!}{3!(26-3)!} = \frac{26!}{3!23!} = \frac{26 \times 25 \times 24}{3 \times 2 \times 1} = 2600 $$
Calculate the number of ways to choose 3 black cards from 26

There are 26 black cards in a standard deck. The number of ways to choose 3 black cards from 26 is:

$$ {26 \choose 3} = \frac{26!}{3!(26-3)!} = \frac{26!}{3!23!} = \frac{26 \times 25 \times 24}{3 \times 2 \times 1} = 2600 $$
Calculate the number of ways to choose 3 red cards and 3 black cards

To find the number of ways to choose 3 red cards and 3 black cards, multiply the number of ways to choose 3 red cards by the number of ways to choose 3 black cards:

$$ {26 \choose 3} \times {26 \choose 3} = 2600 \times 2600 = 6,760,000 $$
Calculate the probability

The probability of choosing 3 red cards and 3 black cards is the number of ways to choose 3 red and 3 black cards divided by the total number of ways to choose 6 cards from the deck:

$$ P(\text{3 red and 3 black}) = \frac{\text{Number of ways to choose 3 red and 3 black}}{\text{Total number of ways to choose 6 cards}} = \frac{6,760,000}{20,358,520} $$
Simplify the probability

Simplify the fraction:

$$ \frac{6,760,000}{20,358,520} = \frac{6760000}{20358520} = \frac{676000}{2035852} = \frac{338000}{1017926} = \frac{169000}{508963} \approx 0.3320 $$
Convert the probability to a fraction close to the options

To express the probability as a simplified fraction that matches the options you provided, we have $\frac{169000}{508963}$. We can approximate this probability as follows:

$\frac{13000}{39151}$

$\frac{13000}{39151}$

More Information

The probability of drawing 3 red cards and 3 black cards when 6 cards are randomly selected from a standard deck of cards is approximately 0.3320 or $\frac{13000}{39151}$.

Tips

A common mistake is not understanding combinations and permutations, specifically when to use each. In this problem, the order in which cards are drawn does not matter, so we are dealing with combinations, not permutations.