Rolling the Dice: Probability Challenge

Solution & Explanation

To solve the problem of calculating the likelihood of rolling at least one '5' in seven dice rolls, we need to understand the probability distribution involved and apply the concept of complementary probability.

Step-by-Step Analysis

1. Understanding the Probability of a Single Roll:

   • When rolling a fair six-sided die, the probability of rolling a '5' is $\frac{1}{6}$.
   • Conversely, the probability of not rolling a '5' (rolling any other number) is $\frac{5}{6}$.

2. Probability of No '5' in Seven Rolls:

   • To find the probability of not rolling a '5' in any of the seven rolls, we use the rule of multiplication for independent events:

     $P(\text{No '5' in 7 rolls}) = \left( \frac{5}{6} \right)^7$

3. Calculating Complementary Probability:

   • The probability of rolling at least one '5' in seven rolls is the complement of the probability of rolling no '5' at all:

     $P(\text{At least one '5' in 7 rolls}) = 1 - P(\text{No '5' in 7 rolls})$

   • Substituting the previous result:

     $P(\text{At least one '5'}) = 1 - \left( \frac{5}{6} \right)^7$

4. Calculating the Numerical Value:

   • Compute the value of $\left( \frac{5}{6} \right)^7$:

     $\left( \frac{5}{6} \right)^7 \approx 0.279$

   • Therefore, the probability of rolling at least one '5' is:

     $P(\text{At least one '5'}) = 1 - 0.279 \approx 0.721$

Conclusion

By using the complementary probability approach, we determined that the probability of rolling at least one '5' in seven dice rolls is approximately 0.721. This method effectively simplifies the problem by focusing on the complementary event, which is often easier to calculate when dealing with "at least" scenarios in probability.