GRE Question of the Day

Answer:

4. 0.3472

The probability of rolling a single 5 on only one out of three dice is \(\frac{5}{6}\times \frac{5}{6}\times \frac{1}{6}\), where \(\frac{5}{6}\) represents the odds of not rolling a 5, while \(\frac{1}{6}\) is the probability of rolling a 5. However, any of the three dice may be the one to roll a 5 so we must add these probabilities together:
 

\(P= (\frac{1}{6}\times \frac{5}{6}\times \frac{5}{6})\)\(+(\frac{5}{6}\times \frac{1}{6}\times \frac{5}{6})\)\(+(\frac{5}{6}\times \frac{5}{6}\times \frac{1}{6})\)

 
This is equivalent to multiplying the odds by 3:
 

\(P=3\times (\frac{5}{6}\times \frac{5}{6}\times \frac{1}{6})\)\(=3\times \frac{25}{216}=\frac{75}{216}\approx 0.347\)

 
This can be thought of as adding the 25 different ways we may roll a single 5 for each die, or 75 ways between all three. This is equal to the number of different ways a single 5 may be rolled with 3 dice (75), and leaves a 34.7% probability of rolling only one 5.