Final answer:
The binomial probability refers to the probability of a specific number of successes in a fixed number of independent Bernoulli trials. The probability of rolling an even number exactly 5 times can be calculated using the binomial probability formula. The probabilities for each scenario are as follows: a. Roll a six-sided number cube 10 times: approximately 0.2461, b. Roll a six-sided number cube 20 times: approximately 0.0148, c. Roll a fair ten-sided die 10 times: approximately 0.2461, d. Roll a fair ten-sided die 20 times: approximately 0.0148.
Step-by-step explanation:
The binomial probability refers to the probability of a specific number of successes in a fixed number of independent Bernoulli trials. In this case, we want to find the probability of rolling an even number exactly 5 times. Let's calculate the probabilities for each scenario:
a. Rolling a six-sided number cube 10 times:
The probability of rolling an even number (2, 4, or 6) on a single roll is 3/6 or 1/2. Since each roll is independent, we can use the binomial probability formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success. In this case, n = 10, k = 5, and p = 1/2. Plugging in these values, we get:
P(X = 5) = (10 choose 5) * (1/2)^5 * (1 - 1/2)^(10-5)
Simplifying the equation, we get P(X = 5) = 252/1024 or approximately 0.2461.
b. Rolling a six-sided number cube 20 times:
Using the same formula, we can calculate the probability for this scenario with n = 20, k = 5, and p = 1/2:
P(X = 5) = (20 choose 5) * (1/2)^5 * (1 - 1/2)^(20-5)
Simplifying the equation gives us P(X = 5) = 15504/1048576 or approximately 0.0148.
c. Rolling a fair ten-sided die 10 times:
The probability of rolling an even number on a single roll is 5/10 or 1/2. Using the binomial probability formula with n = 10, k = 5, and p = 1/2:
P(X = 5) = (10 choose 5) * (1/2)^5 * (1 - 1/2)^(10-5)
Simplifying the equation gives us P(X = 5) = 252/1024 or approximately 0.2461.
d. Rolling a fair ten-sided die 20 times:
Using the same formula, we can calculate the probability for this scenario with n = 20, k = 5, and p = 1/2:
P(X = 5) = (20 choose 5) * (1/2)^5 * (1 - 1/2)^(20-5)
Simplifying the equation gives us P(X = 5) = 15504/1048576 or approximately 0.0148.