N= 80 p = 0.7 x = 62?

A) Exact: 0.0335; Approximate: 0.0334
B) Exact: 0.0343; Approximate 0.0334
C) Exact: 0.0343; Approximate: 0.0350
D) Exact: 0.0343; Approximate: 0.0342
Which one is the correct answer and can you show work please?

Answer is D. Use a Calculator.

To find the answer to this question, we need to calculate the binomial probability. The formula for the binomial probability is:

P(x) = (nCx) * p^x * (1 - p)^(n - x)

Where:
- n is the number of trials or observations (in this case, n = 80)
- x is the number of successes (in this case, x = 62)
- p is the probability of success on a single trial (in this case, p = 0.7)
- (nCx) is the number of ways to choose x items from a set of n items, which can be calculated as n! / (x! * (n - x)!), where ! denotes factorial.

Let's calculate the exact and approximate values using the given values:

For the exact value:
P(x) = (80C62) * 0.7^62 * (1 - 0.7)^(80 - 62)

To calculate (80C62), we can use the formula:
(80C62) = 80! / (62! * (80 - 62)!)

For the approximate value:
P(x) = (80C62) * 0.7^62 * (1 - 0.7)^(80 - 62)

Now, in order to compute (80C62), we first need to calculate 80! (Factorial of 80). The factorial of a number is the product of all positive integers less than or equal to that number. However, calculating 80! directly can be quite time-consuming, so we can use an approximation formula called Stirling's Approximation:

n! ≈ √(2πn) * (n/e)^n

By using Stirling's Approximation, we can compute an approximate value for 80! without having to calculate the factorial step by step.

Let's calculate the exact and approximate values:

Exact Value:
P(x) = (80C62) * 0.7^62 * (1 - 0.7)^(80 - 62)
= [80! / (62! * (80 - 62)!)] * 0.7^62 * 0.3^18

Approximate Value:
P(x) = (√(2π80) * (80/e)^80) / (√(2π62) * (62/e)^62 * √(2π18) * (18/e)^18) * 0.7^62 * 0.3^18

Now, we can use a calculator to evaluate these expressions. Comparing the answers to the given options:

A) Exact: 0.0335; Approximate: 0.0334
B) Exact: 0.0343; Approximate 0.0334
C) Exact: 0.0343; Approximate: 0.0350
D) Exact: 0.0343; Approximate: 0.0342

We can see that the correct answer is:

C) Exact: 0.0343; Approximate: 0.0350

Please note that the exact value has been rounded to four decimal places, so there might be some small differences due to rounding.