Questions: Determine whether the scenario below can be modeled with a binomial distribution: You keep rolling a die until you roll either a 3 or a 6.

Transcript text: Determine whether the scenario below can be modeled with a binomial distribution: You keep rolling a die until you roll either a 3 or a 6.

Solution

Solution Steps

Step 1: Determine Fixed \( n \)

In the given scenario, we keep rolling a die until we roll either a 3 or a 6. Since we do not have a predetermined number of trials (the number of rolls can vary), we conclude that \( n \) is not fixed.

Step 2: Assess Independence

Each roll of the die is independent of the previous rolls. The outcome of one roll does not affect the outcome of another. Therefore, the trials are independent.

Step 3: Check for Binary Outcomes

In this scenario, there are two possible outcomes for each roll:

Success: rolling a 3 or a 6.
Failure: rolling any other number (1, 2, 4, or 5).

Thus, the scenario meets the criterion for binary outcomes.

Step 4: Evaluate Same Probability \( p \)

The probability of success (rolling a 3 or 6) remains constant for each roll. The probability can be calculated as follows: \[ p = P(\text{rolling a 3 or 6}) = \frac{2}{6} = \frac{1}{3} \] The probability of failure \( q \) is: \[ q = 1 - p = 1 - \frac{1}{3} = \frac{2}{3} \]

Step 5: Conclusion on Binomial Distribution

Despite having independent trials and binary outcomes with a constant probability of success, the lack of a fixed number of trials means that this scenario cannot be modeled with a binomial distribution.