help please
There are 12 red checkers and 3 black checkers in a bag. Checkers are selected one at a time, with replacement. Each time, the color of the checker is recorded. Find the probability of selecting a red checker exactly 7 times in 10 selections. Show work.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
(12/15)^7 * (3/15)^3 = ?
so the answer is 0.0017?
To find the probability of selecting a red checker exactly 7 times in 10 selections, we need to determine the number of ways this can occur and divide it by the total number of possible outcomes.
We'll use the binomial probability formula, which is given by: P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) represents the probability of getting exactly k successes.
C(n, k) represents the number of combinations of n items taken k at a time.
p represents the probability of success on a single trial.
n represents the total number of trials.
Let's calculate the probability step-by-step:
Step 1: Determine the number of ways to select exactly 7 red checkers in 10 selections.
We can use the combination formula to calculate this:
C(10, 7) = 10! / (7! * (10 - 7)!)
= (10 * 9 * 8) / (3 * 2 * 1)
= 120
So there are 120 ways to select exactly 7 red checkers.
Step 2: Calculate the probability of getting a red checker on a single trial.
Since there are 12 red checkers and 15 total checkers (12 red + 3 black), the probability of getting a red checker is:
p = 12 / 15 = 4 / 5
Step 3: Calculate the probability of selecting a red checker exactly 7 times in 10 selections using the binomial probability formula:
P(X = 7) = C(10, 7) * (4/5)^7 * (1 - 4/5)^(10 - 7)
= 120 * (4/5)^7 * (1/5)^3
P(X = 7) = 120 * (0.3277) * (0.008)
P(X = 7) ≈ 0.314
Therefore, the probability of selecting a red checker exactly 7 times in 10 selections is approximately 0.314, or 31.4%.