To find the probability of Sandra randomly selecting a $10 bill and then a $50 bill from the envelope, we'll first determine the total number of bills in the envelope and then calculate the probabilities step by step.
-
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 bills.
-
Calculate the probability of selecting a $10 bill first: The probability of selecting a $10 bill on the first draw is the number of $10 bills divided by the total number of bills: \[ P(\text{first bill is } $10) = \frac{5}{30} = \frac{1}{6} \]
-
Calculate the probability of selecting a $50 bill second: After drawing a $10 bill, there will be:
- Total bills left = 30 - 1 = 29 bills,
- Number of $50 bills left = 15 (no $50 bill has been drawn yet).
Thus, the probability of selecting a $50 bill on the second draw is: \[ P(\text{second bill is } $50 | \text{first bill was } $10) = \frac{15}{29} \]
-
Calculate the combined probability: The combined probability of both events happening (first selecting a $10 bill and then selecting a $50 bill) is: \[ P(\text{first is } $10 \text{ and second is } $50) = P(\text{first is } $10) \times P(\text{second is } $50 | \text{first is } $10) \] \[ = \frac{1}{6} \times \frac{15}{29} = \frac{15}{174} \]
-
Simplify the fraction: Now we simplify \(\frac{15}{174}\). The greatest common divisor of 15 and 174 is 3: \[ \frac{15 \div 3}{174 \div 3} = \frac{5}{58} \]
Therefore, the probability of Sandra selecting a $10 bill followed by a $50 bill is \(\boxed{\frac{5}{58}}\).