Understanding the Probability of Rolling a 4 at Least Once in 4 Dies

In the realm of probability, understanding the likelihood of events can provide valuable insights into random outcomes. One common problem involves calculating the probability of rolling a specific number at least once in a series of die rolls. In this case, we will explore the probability of rolling a 4 at least once when a die is rolled 4 times.

The Problem

The problem is to determine the probability of rolling the number 4 at least once when rolling a die 4 times. We will break this down step by step, using both combinatorial methods and direct probability calculations.

Combinatorial Approach

To solve this problem, we will use combinatorial methods to determine the total number of favorable outcomes and the total possible outcomes. Total possible outcomes: 6^4 1296 Number of ways to roll no 4s: 5^4 625 Number of ways to roll at least one 4: 1296 - 625 671 From this, we can calculate the probability:

P(at least one 4) 671 / 1296 ≈ 0.5177

Thus, the probability of rolling a 4 at least once in 4 rolls is approximately 51.77%.

Direct Probability Calculation

Another way to approach this problem is by calculating the probability of the complementary event, which is the event that a 4 is not rolled at all in 4 rolls. The probability of not rolling a 4 on a single roll is 5/6. Therefore, the probability of not rolling a 4 in all 4 rolls is:

P(no 4s) (5/6)^4 625/1296

The probability of rolling a 4 at least once is then the complement of this event:

P(at least one 4) 1 - P(no 4s) 1 - 625/1296 671/1296 ≈ 0.5177

Additional Scenarios

Let's consider the probability of rolling at least two 4s in four rolls of a die. We can use the inclusion-exclusion principle for this calculation.

Probability of rolling no 4s: (5/6)^4 625/1296

Probability of rolling exactly one 4: 4 * (1/6) * (5/6)^3 4 * 125/1296 500/1296

Therefore, the probability of rolling at least two 4s: 1 - (625/1296 500/1296) 1 - 1125/1296 171/1296 ≈ 0.131944444

Conclusion

In conclusion, the probability of rolling a 4 at least once in 4 rolls of a die is approximately 51.77%, or 671/1296 when expressed as a fraction. Similarly, the probability of rolling a 4 at least twice is approximately 13.19%, or 171/1296. These calculations highlight the importance of both the direct method and the use of complementary events in solving probability problems.

Keywords: probability, die roll, at least one 4