Final answer:
The question requires calculating a binomial probability for n=8, x=5, and p=0.8 by finding the number of combinations and multiplying by the probabilities of success and failure.
Step-by-step explanation:
The student is asking to evaluate a binomial probability expression C ₙ, ₓp ˣq ⁿₓ given specific values for n, x, and p. For n=8, x=5, and p=0.8, we can calculate the probability using the binomial probability formula, which is P(x) = C(n, x) * pˣ * qⁿₓ, where q is the probability of failure (q=1-p). The coefficient C(n, x) represents the number of combinations of n things taken x at a time, often written as nCx or ℂ(ₙ, ₓ).
To solve for C(8, 5), we need to calculate the number of combinations of 8 things taken 5 at a time. Then we multiply this number by 0.8³ for the probability of 5 successes and (1-0.8)³ for the probability of 3 failures.
The number of combinations is calculated using the factorial function: C(8,5) = 8! / (5!(8 - 5)!). Plugging in the values, we get C(8,5) = 8! / (5!3!) = (8 × 7 × 6) / (3 × 2 × 1) = 56. Now we calculate the probabilities: pˣ = 0.8³ = 0.32768 and qⁿₓ = 0.2³ = 0.008. Multiplying these with the combination count gives us P(x=5) = 56 × 0.32768 × 0.008, which we can then compute.