Final answer:
To calculate the probability of rolling an even number exactly 5 times when rolling a six-sided die 10 times, use the binomial probability formula. So the correct option is b.
Step-by-step explanation:
To find the probability that you will roll an even number exactly 5 times when you roll a six-sided number cube 10 times, you should use the binomial probability formula. In this situation, each roll of the die represents a Bernoulli trial with two possible outcomes - rolling an even number (success) or not (failure).
The probability of success (rolling an even number) is 1/2 since there are three even numbers (2, 4, 6) out of six possible outcomes.
To calculate the exact probability of 5 successes (even numbers) in 10 trials, use the binomial probability formula:
P(X = k) = C(n, k) × p^k × (1-p)^(n-k)
Where:
C(n, k) is the combination of n items taken k at a time.
p is the probability of a single success (1/2 for an even number).
n is the number of trials (10 in this case).
k is the number of successes (5 in this case).
Using this formula, calculate C(10, 5), which is the number of ways to choose 5 successes out of 10 trials.
Then, multiply this by the probability of getting 5 even numbers raised to the 5th power, and the probability of not getting an even number (1-1/2 or 1/2) raised to the 5th power since there are 5 trials where an even number is not rolled.
The resulting number is the probability of exactly 5 even numbers in 10 rolls of the die.
So the correct option is b.