Question

we draw three cards from a full deck. what is the probability that they are all red? (enter the probability in decimal form rounded to four places.) the probability they are all red is

Explanation:

Step1: Calculate the total number of ways to draw 3 cards

The total number of ways to draw 3 cards from a deck of 52 cards is given by the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 52\) and \(r=3\).

$$C(52,3)=\frac{52!}{3!(52 - 3)!}=\frac{52\times51\times50}{3\times2\times1}=22100$$

Step2: Calculate the number of ways to draw 3 red cards

There are 26 red cards in a deck. The number of ways to draw 3 red cards is \(C(26,3)\) with \(n = 26\) and \(r = 3\).

$$C(26,3)=\frac{26!}{3!(26- 3)!}=\frac{26\times25\times24}{3\times2\times1}=2600$$

Step3: Calculate the probability

The probability \(P\) that all three cards are red is the number of favorable outcomes (drawing 3 red cards) divided by the number of total outcomes (drawing any 3 cards).

$$P=\frac{C(26,3)}{C(52,3)}=\frac{2600}{22100}\approx0.1176$$

Answer:

\(0.1176\)