To determine the standard deviation for the number of times that a 3 will be rolled when a die is rolled 360 times, we start by recognizing that this scenario can be modeled using a binomial distribution.
In a binomial distribution, we have:
- n: the number of trials (rolls of the die), which is 360 in this case.
- p: the probability of success on each trial (rolling a 3). Since there are 6 faces on a die, the probability of rolling a 3 is
1/6.
Next, we calculate the expected number of successes (3s rolled), which is given by:
Expected hits (mean, μ) = n * p
μ = 360 * (1/6) = 60
Now, for a binomial distribution, the standard deviation (σ) is calculated as:
Standard Deviation (σ) = sqrt(n * p * (1 – p))
Substituting the values we have:
σ = sqrt(360 * (1/6) * (5/6))
σ = sqrt(360 * (5/36))
σ = sqrt(50)
σ ≈ 7.07
Thus, the standard deviation for the number of times a 3 is rolled when a die is rolled 360 times is approximately 7.07. This means that you can expect some variation around the average of 60, with about 68% of the rolls falling within one standard deviation of that mean (between 53 and 67 when rounded).