Let's solve this problem step-by-step to find the probability of drawing exactly 1 ace when two cards are drawn from a well-shuffled deck of 52 playing cards.
### Step 1: Understand the Problem
We are given:
- A standard deck contains 52 playing cards.
- Among these cards, there are 4 aces.
- We draw 2 cards from the deck.
- We need to find the probability of getting exactly 1 ace in those 2 cards.
### Step 2: Define the Problem
Let's denote the number of aces drawn by [tex]\(X\)[/tex]. We need to find [tex]\(P(X=1)\)[/tex], which is the probability that exactly 1 ace is drawn.
### Step 3: Calculate the Total Number of Ways to Draw 2 Cards
The total number of ways to draw 2 cards out of 52 can be calculated using the combination formula:
[tex]\[
\binom{52}{2} = \frac{52!}{2!(52-2)!} = \frac{52 \times 51}{2 \times 1} = 1326
\][/tex]
### Step 4: Calculate the Number of Favorable Outcomes
To draw exactly 1 ace and 1 non-ace, we need:
- The number of ways to choose 1 ace from the 4 available aces, which can be calculated using the combination formula:
[tex]\[
\binom{4}{1} = 4
\][/tex]
- The number of ways to choose 1 non-ace from the remaining 48 non-aces (52 total cards minus 4 aces), which can be calculated using the combination formula:
[tex]\[
\binom{48}{1} = 48
\][/tex]
### Step 5: Multiply the Number of Favorable Outcomes
The total number of favorable outcomes is the product of the two combinations calculated in Step 4:
[tex]\[
\binom{4}{1} \times \binom{48}{1} = 4 \times 48 = 192
\][/tex]
### Step 6: Calculate the Probability
The probability [tex]\(P(X=1)\)[/tex] of drawing exactly 1 ace when 2 cards are drawn is the ratio of the number of favorable outcomes to the total number of possible outcomes:
[tex]\[
P(X=1) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{192}{1326}
\][/tex]
### Step 7: Simplify the Result (if needed)
Upon simplifying or calculating this fraction, we get:
[tex]\[
P(X=1) \approx 0.14479638009049775
\][/tex]
So, the probability of drawing exactly 1 ace when 2 cards are drawn from a well-shuffled deck of 52 cards is approximately [tex]\(0.1448\)[/tex] or [tex]\(14.48\%\)[/tex].
