Final answer:
The probability of rolling exactly one odd number with a fair die in ten rolls is represented by the binomial probability formula, resulting in 10(1/2)^10 (option 2) or approximately 0.00977.
Step-by-step explanation:
The student is interested in the probability of obtaining exactly one odd number when a fair six-sided die is rolled ten times.
In probabilistic terms, the random variable X describes the number of times the resulting face is odd. To find the probability of X=1, we recognize that each die roll is an independent event, with the probability of getting an odd number (1, 3, or 5) being 1/2, since there are three odd faces out of six. Applying the concept of binomial probability, we use the binomial formula:
P(X=k) = C(n, k) × (p)^k × (q)^(n-k)
Where P(X=k) is the probability of k successes in n trials, C(n, k) is the binomial coefficient representing the number of combinations of n items taken k at a time, p is the probability of success on a single trial, and q is the probability of failure on a single trial.
For this situation, n = 10 (number of rolls), k = 1 (one odd number), p = 1/2 (success probability), and q = 1/2 (failure probability). Plugging these into the formula gives:
P(X=1) = C(10, 1) × (1/2)^1 × (1/2)^(10-1)
Calculating this, we get:
P(X=1) = 10 × (1/2)^1 × (1/2)^9
P(X=1) = 10 × (1/2)^10
P(X=1) = 10/1024, or approximately 0.00977
So, the correct answer is 10(1/2)¹⁰, which represents the probability of obtaining exactly one odd number when the die is rolled ten times.