Final answer:
A binomial test is used to assess whether a die's performance aligns with the expected probability. For a fair die, rolling a '5' should occur approximately 1/6 of the time. By applying the binomial formula, we can calculate the probability of getting the observed value, or more extreme, compared to the expected frequency.
Step-by-step explanation:
To determine if a die's performance aligns with the expected probability, we can conduct a Binomial test. With a fair six-sided die, each side should land approximately 1/6 of the time when rolled a large number of times. In this scenario, rolling a '5' has a theoretical probability of 1/6, and we rolled the die a total of 24 times.
First, we set up the null hypothesis which assumes the die is fair. So, the expected number of times to roll a '5' would be 24 rolls multiplied by the probability of a single '5', which is 1/6:
Expected frequency = 24 * (1/6) = 4
However, the die landed on '5' a total of 20 times. To assess whether this result significantly deviates from the expectation, we can use the binomial formula:
P(X = k) = (n choose k) * (p)^k * (1-p)^(n-k)
Where,
P(X = k) is the probability of getting k successes in n trials
n is the number of trials
k is the number of successes
p is the probability of success on an individual trial
Using this formula, we could calculate the probability of getting exactly 20 '5's out of 24 rolls, and also account for the probabilities of getting any number from 5 to 24 '5's to determine if our observed frequency is significantly higher than expected by chance.