To find the probability of Sandra selecting a $10 bill first and then a $50 bill, we can follow these steps:
-
Calculate the total number of bills in the envelope:
- Number of $10 bills: 5
- Number of $20 bills: 10
- Number of $50 bills: 15
Total number of bills = \( 5 + 10 + 15 = 30 \)
-
Calculate the probability of selecting a $10 bill first:
- The probability of selecting a $10 bill = \( \frac{\text{Number of $10 bills}}{\text{Total number of bills}} = \frac{5}{30} = \frac{1}{6} \)
-
Calculate the probability of selecting a $50 bill second, given that a $10 bill has already been selected (which reduces the total count by 1):
- After selecting a $10 bill, the number of remaining bills = \( 30 - 1 = 29 \)
- The number of $50 bills remains the same at 15.
- The probability of selecting a $50 bill now = \( \frac{\text{Number of $50 bills}}{\text{New total number of bills}} = \frac{15}{29} \)
-
Calculate the combined probability of both events:
- Total probability = Probability of first event × Probability of second event
- \( P(\text{10 then 50}) = \left(\frac{5}{30}\right) \times \left(\frac{15}{29}\right) = \left(\frac{1}{6}\right) \times \left(\frac{15}{29}\right) = \frac{15}{174} \)
-
Simplify the fraction:
- The fraction \( \frac{15}{174} \) can be simplified by finding the greatest common divisor (GCD) of 15 and 174, which is 3.
- Dividing both the numerator and denominator by 3:
- \( \frac{15 \div 3}{174 \div 3} = \frac{5}{58} \)
Thus, the probability of getting a $10 bill first and then a $50 bill is \( \frac{5}{58} \).