Calculating the Probability of Rolling a 3 or 6 for the First Time on the Sixth Roll

Consider a scenario where you roll an unbiased six-sided die 40 times and observe 13 outcomes as either a 3 or a 6. While this historical data might be interesting, it doesn't influence the probability of future rolls. Given this context, let's explore the specific problem of determining the probability of rolling a 3 or 6 for the first time on the sixth roll of a fair die.

Understanding the Problem

In an unbiased game of dice, the probability of achieving any specific outcome on a given roll is the same. Therefore, the probability of rolling a 3 or a 6 on a single roll is straightforward:

Probability of Success (Rolling a 3 or 6)

The probability of rolling a 3 or 6 on a single roll can be calculated as follows:

Let P3 or 6 represent the probability of rolling a 3 or 6:

P3 or 6 P(3) P(6) 1/6 1/6 2/6 1/3

Probability of Failure (Not Rolling a 3 or 6)

Conversely, the probability of not rolling a 3 or 6 on a single roll is:

P(not 3 or 6) 1 - P(3 or 6) 1 - 1/3 2/3

Geometric Distribution and Probability of First Success on the Sixth Roll

To determine the probability of getting a 3 or 6 for the first time on the sixth roll, we need to consider that the first five rolls must be failures (not 3 or 6), and the sixth roll must be a success (3 or 6).

Probability of First Success on the Sixth Roll

The probability of the first success on the sixth roll can be calculated using the geometric distribution formula:

P(first success on 6th roll) P(not 3 or 6)5 × P(3 or 6)

Calculating the Probability

Substitute the probabilities:

P(first success on 6th roll) (2/3)5 × 1/3

Calculate (2/3)5:

(2/3)5 25 / 35 32 / 243

Then multiply by 1/3:

P(first success on 6th roll) 32 / 243 × 1 / 3 32 / 729

Final Probability

The probability of getting a 3 or 6 for the first time on the sixth roll is:

32/729

Expressed in LaTeX format, the final answer is:

boxed{frac{32}{729}}

Revisiting the Initial Information

It's important to note that the historical data (40 rolls and 13 successes) doesn't influence the current roll probabilities. Each roll is an independent event, so the probability of not getting a 3 or 6 in the first five rolls, and successfully rolling a 3 or 6 on the sixth roll, is the same regardless of previous outcomes:

Probability of Not Getting a 3 or 6 in the First Five Rolls

The probability of not getting a 3 or 6 in any of the first five rolls is:

(2/3) × (2/3) × (2/3) × (2/3) × (2/3) (2/3)5 32 / 243

Probability of Getting a 3 or 6 on the Sixth Roll

The probability of getting a 3 or 6 on the sixth roll, if the previous five rolls were also not 3 or 6, is:

1/3

Thus, the total probability of these two events occurring in sequence is:

(32 / 243) × (1 / 3) 32 / 729