Question

Assume that a procedure yields a binomial distribution with a trial repeated n = 30 times. Use the binomial probability formula to find the probability of x = 5 successes given the probability p = 1/5 of success on a single trial. Round to three decimal places. O 0.172 0.198 O 0.421 O 0.067

Answer 1

The question asks for the probability of getting exactly 5 successes in 30 binomial trials with a success probability of 1/5, which can be found using the binomial probability formula and involves calculating combinations and raising probabilities to the respective powers.

Step-by-step explanation:

The probability of exactly x = 5 successes in n = 30 trials when the probability of success on a single trial is p = 1/5 can be found using the binomial probability formula:

P(X = x) = (n choose x) * p^x * q^(n - x), where q = 1 - p is the probability of failure on a single trial.

First, calculate q:
q = 1 - p
q = 1 - (1/5)
q = 4/5

Then, use the combination formula for (n choose x):
(30 choose 5) = 30! / (5! * (30 - 5)!)

Next, plug in values into the binomial formula:
P(X = 5) = (30 choose 5) * (1/5)^5 * (4/5)^(30 - 5)

Solving the above expression gives us the probability of 5 successes in 30 trials with a success probability of 1/5. Remember to round to three decimal places as requested.

Answer 2

The probability of x = 5 successes is approximately 0.186.

To find the probability of x = 5 successes in a binomial distribution with n = 30 trials and a probability of success p = 1/5 on a single trial, we can use the binomial probability formula.

The binomial probability formula is:

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

where nCx represents the number of combinations of n items taken x at a time.

In this case, we want to find P(x = 5). Plugging in the given values, we get:

P(5) = (30C5) * ((1/5)^5) * ((4/5)^(30-5))

Now, let's calculate each part step by step:

1. (30C5): This represents the number of combinations of 30 items taken 5 at a time. We can calculate it using the formula:

(30C5) = 30! / (5! * (30-5)!)

Using this formula, we find that (30C5) = 142506.

2. (1/5)^5: This represents the probability of success (1/5) raised to the power of 5.

(1/5)^5 = 1/3125 ≈ 0.00032 (rounded to five decimal places).

3. (4/5)^(30-5): This represents the probability of failure (4/5) raised to the power of (n - x), where n is the number of trials and x is the number of successes.

(4/5)^(30-5) = (4/5)^25 ≈ 0.00389 (rounded to five decimal places).

Now, let's multiply all the parts together:

P(5) = 142506 * 0.00032 * 0.00389 ≈ 0.186

Rounding this to three decimal places, the probability of x = 5 successes is approximately 0.186.

Therefore, the correct answer is 0.186.