A bag contains 38 U.S. quarters and twelve Canadian quarters. (The coins are identical in size.) If four quarters are randomly picked from the bag, what is the probability of getting at least one Canadian quarter? (Round your answer to one decimal place.)

To solve this problem, we need to calculate the probability of getting at least one Canadian quarter when four quarters are randomly picked from a bag containing U.S. and Canadian quarters.

### Step-by-Step Solution:

1. **Total Number of Quarters:**
   - U.S. quarters: 38
   - Canadian quarters: 12
   - Total quarters = \(38 + 12 = 50\)

2. **Finding the Probability of the Complement Event:**
   - The complement event to "getting at least one Canadian quarter" is "getting no Canadian quarters" (i.e., all four quarters picked are U.S. quarters).

3. **Calculating the Number of Ways to Choose Quarters:**
   - **Total ways to pick 4 quarters out of 50:**  
     \[
     \binom{50}{4} = \frac{50!}{4!(50-4)!}
     \]
   - **Ways to pick 4 U.S. quarters out of 38:**  
     \[
     \binom{38}{4} = \frac{38!}{4!(38-4)!}
     \]

4. **Calculate Each Combination:**
   - \(\binom{50}{4} = \frac{50 \times 49 \times 48 \times 47}{4 \times 3 \times 2 \times 1} = 230300\)
   - \(\binom{38}{4} = \frac{38 \times 37 \times 36 \times 35}{4 \times 3 \times 2 \times 1} = 73815\)

5. **Probability of Picking No Canadian Quarters:**
   \[
   P(\text{no Canadian quarters}) = \frac{\binom{38}{4}}{\binom{50}{4}} = \frac{73815}{230300} \approx 0.3205
   \]

6. **Probability of Getting At Least One Canadian Quarter:**
   \[
   P(\text{at least one Canadian quarter}) = 1 - P(\text{no Canadian quarters}) = 1 - 0.3205 = 0.6795
   \]

7. **Rounding the Result:**
   - Rounding \(0.6795\) to one decimal place, we get \(0.7\).

### Conclusion:
The probability of getting at least one Canadian quarter when four quarters are randomly picked from the bag is **0.7** (rounded to one decimal place).

---

Would you like to see more details or have any questions?

Here are 5 related questions to consider:
1. What is the probability of getting exactly two Canadian quarters?
2. How would the probability change if there were more Canadian quarters?
3. What if the quarters are picked one by one with replacement? How does this affect the probability?
4. Can you calculate the expected number of Canadian quarters in four picks?
5. How would the result change if instead of four, six quarters were picked?

**Tip:** When dealing with probabilities, it's often easier to calculate the complement and subtract from 1, especially when dealing with "at least one" scenarios.

Learn how to calculate the probability of picking at least one Canadian quarter from a bag containing U.S. and Canadian quarters. Step-by-step solution using combinatorics and probability theory provided. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation