In basketball, Nicole makes 4 baskets for every 10 shots. If she takes 3 shots, what is the probability that exactly 2 of them will be baskets?
4/10 * 4/10 * 6/10 = ?
To find the probability of Nicole making exactly 2 baskets out of 3 shots, we can use binomial probability.
Binomial probability calculates the probability of getting a certain number of successes (in this case, making a basket) in a fixed number of Bernoulli trials (in this case, taking shots), given a specific probability of success on each trial (in this case, the probability of making a basket).
The formula to calculate binomial probability is:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of getting exactly k successes,
- C(n, k) is the binomial coefficient, which calculates the number of combinations for k successes out of n trials,
- p is the probability of success on a single trial, and
- n is the total number of trials.
In this case, Nicole makes 4 baskets out of every 10 shots, so the probability of success (p) is 4/10 or 2/5. The total number of trials (n) is 3.
Let's calculate the probability:
P(X = 2) = C(3, 2) * (2/5)^2 * (3/5)^(3 - 2)
The binomial coefficient C(3, 2) is calculated as follows:
C(3, 2) = 3! / (2! * (3 - 2)!) = 3
Substituting the values into the formula:
P(X = 2) = 3 * (2/5)^2 * (3/5)^(3 - 2)
= 3 * (4/25) * (3/5)^1
= 3 * 4/25 * 3/5
= 12/25
Therefore, the probability that Nicole will make exactly 2 baskets out of 3 shots is 12/25.