mike has been shooting paper wads at huis trash can all morning making 60% of them. if he throws 3 more paper wads, what is the probability that he will make exactly two of the three shots?

To calculate the probability that Mike will make exactly two out of the three additional paper wad shots, we use the **binomial probability formula**:

\[
P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}
\]

Where:
- \( n \) = total number of trials (in this case, 3),
- \( k \) = number of successes (in this case, 2),
- \( p \) = probability of success on a single trial (60% = 0.6),
- \( \binom{n}{k} \) = the number of ways to choose \( k \) successes from \( n \) trials, calculated as:

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

### Step-by-Step Solution:

1. **Calculate \(\binom{n}{k}\):**
   \[
   \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \cdot 2 \cdot 1}{2 \cdot 1 \cdot 1} = 3
   \]

2. **Substitute into the binomial formula:**
   \[
   P(X = 2) = \binom{3}{2} \cdot p^2 \cdot (1-p)^{3-2}
   \]
   Substituting \( p = 0.6 \):
   \[
   P(X = 2) = 3 \cdot (0.6)^2 \cdot (1-0.6)^1
   \]

3. **Simplify the terms:**
   - \( (0.6)^2 = 0.36 \),
   - \( 1-0.6 = 0.4 \),
   - \( 0.4^1 = 0.4 \).

   So:
   \[
   P(X = 2) = 3 \cdot 0.36 \cdot 0.4
   \]

4. **Calculate the result:**
   \[
   P(X = 2) = 3 \cdot 0.144 = 0.432
   \]

### Final Answer:
The probability that Mike will make exactly 2 out of the 3 additional shots is **0.432**, or **43.2%**.

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Would you like a detailed explanation of the binomial formula or its applications? 😊

### Related Questions:
1. What is the probability Mike will make all 3 shots?
2. How does the probability change if Mike’s success rate improves to 70%?
3. What is the expected number of successful shots out of 3?
4. How would the probability distribution for 3 shots look graphically?
5. What is the variance and standard deviation for Mike’s performance over 3 shots?

**Tip:** The binomial formula is powerful for solving discrete probability problems involving repeated independent trials!

Mike has been shooting paper wads at his trash can all morning making 60% of them. If he throws 3 more paper wads, what is the probability that he will make exactly two of the three shots?

Calculate the probability of making exactly two out of three paper wad shots using the binomial probability formula. This problem applies to discrete probability and binomial distribution, with a 60% success rate for each shot. Suitable for Grades 9-12.

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