Question

A money envelope contains five $10
bills, ten $20
bills, and fifteen $50
bills. Sandra randomly selects two bills without returning the first bill. What is the probability of getting a $10
bill then a $50
bill? Write your answer in the simplest form of fraction.(1 point)

Answer 1

To find the probability of selecting a $10 bill followed by a $50 bill without replacement, we can follow these steps:

1. **Count the Total Number of Bills**:
- Number of $10 bills = 5
- Number of $20 bills = 10
- Number of $50 bills = 15
- Total number of bills = 5 + 10 + 15 = 30

2. **Calculate the Probability of Selecting a $10 Bill First**:
- Probability of selecting a $10 bill = Number of $10 bills / Total number of bills
\[
P(10) = \frac{5}{30} = \frac{1}{6}
\]

3. **Update the Total Number of Bills After Selecting a $10 Bill**:
- After selecting one $10 bill, the remaining number of bills = 30 - 1 = 29

4. **Calculate the Probability of Selecting a $50 Bill Second**:
- Number of $50 bills remains 15 (since we selected a $10 bill).
- Probability of selecting a $50 bill now = Number of $50 bills / Remaining total number of bills
\[
P(50 | 10) = \frac{15}{29}
\]

5. **Compute the Joint Probability of Both Events**:
- The total probability of selecting a $10 bill followed by a $50 bill is the product of the two probabilities:
\[
P(10 \text{ and then } 50) = P(10) \cdot P(50 | 10) = \frac{1}{6} \cdot \frac{15}{29}
\]

Multiplying the fractions:
\[
P(10 \text{ and then } 50) = \frac{1 \cdot 15}{6 \cdot 29} = \frac{15}{174}
\]

6. **Simplify the Fraction**:
- Now we simplify \(\frac{15}{174}\). The greatest common divisor (GCD) of 15 and 174 is 3.
\[
\frac{15 \div 3}{174 \div 3} = \frac{5}{58}
\]

Thus, the probability of getting a $10 bill followed by a $50 bill is \(\boxed{\frac{5}{58}}\).