Question

Assume that a procedure yields a binomial distribution with a trial repeated n = 12 times. Use either the binomial probability formula (or a technology like Excel or StatDisk) to find the probability of k = 9 3 successes given the probability p 3/10 of success on a single trial. 10 (Report answer accurate to 4 decimal places.) P(X = k) =

Answer 1

The probability of getting exactly 9 successes in a binomial experiment with 12 trials and a success probability of 3/10 is found using the binomial probability formula. Substituting the values, the probability is calculated to be approximately 0.0002 or reported to four decimal places as 0.0002.

Step-by-step explanation:

To find the probability of exactly 9 successes (k = 9) in 12 trials (n = 12) of a binomial experiment with a success probability of 3/10 (p = 0.3), we use the binomial probability formula:

P(X = k) = (n choose k) * p^k * q^(n-k), where q is the probability of failure and equals 1 - p.

Plugging in the values, we get:

P(X = 9) = (12 choose 9) * (0.3)^9 * (0.7)^3

First, calculate (12 choose 9), which is the combination of 12 items taken 9 at a time. This equals 220. Now, raise 0.3 to the 9th power and 0.7 to the 3rd power, then multiply these with 220.

P(X = 9) ≈ 220 * (0.3)^9 * (0.7)^3 ≈ 0.0002

Therefore, the probability of getting exactly 9 successes is approximately 0.0002, or with four decimal places, 0.0002.

Answer 2

To find the probability of getting exactly 9 successes out of 12 trials with a probability of success of 3/10, you can use the binomial probability formula.

Step-by-step explanation:

To find the probability of getting exactly 9 successes out of 12 trials with a probability of success of 3/10, we can use the binomial probability formula.

The formula is P(X = k) = nCk * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the probability of success, and nCk represents the number of combinations of n items taken k at a time.

Plugging in the values, we have n = 12, k = 9, p = 3/10.

Using the combination formula, nCk = 12C9 = 12! / (9! * (12-9)!) = 220.

So, P(X = 9) = 220 * (3/10)^9 * (1-(3/10))^(12-9).

Calculating this, we get P(X = 9) ≈ 0.010206, rounded to 4 decimal places.