Final answer:
The standard deviation for rolling a 1 or 2 on a fair six-sided die 36 times is 2.828, calculated using the binomial standard deviation formula with a probability of success of 1/3.
Step-by-step explanation:
The student is asking about the standard deviation for obtaining a 1 or 2 on a six-sided die when rolled 36 times. Since each roll is independent, we can treat this as a binomial distribution problem, where the probability of success (rolling a 1 or 2) on a single trial (roll of the die) is P(success) = 1/3, as two out of the six possible outcomes are considered a success.
To find the standard deviation for a binomial distribution, we use the formula σ = √(n × p × (1-p)), where σ is the standard deviation, n is the number of trials, p is the probability of success on a single trial, and (1-p) is the probability of failure.
So, for 36 rolls:
- Calculate the probability of success: p = 1/3 (since there are two favorable outcomes, 1 and 2, out of six total outcomes)
- Calculate the number of trials: n = 36
- Use the binomial standard deviation formula: σ = √(36 × (1/3) × (2/3))
- Solve for σ: σ = √(36 × 1/3 × 2/3) = √(36 × 1/3 × 2/3) = √(8) = 2.828
Therefore, the standard deviation for the number of times a 1 or 2 appears on the top face after rolling a fair die 36 times is 2.828.