a basket ball player makes 65% of her free throws find the probability that she will make at least 7 of her next 8 free throws

To find the probability that the basketball player will make at least 7 of her next 8 free throws, we can use the binomial probability formula.

The binomial probability formula is given by:

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

Where:
- P(x) is the probability of getting exactly x successes
- n is the number of trials (in this case, the number of free throws)
- x is the number of successes (the number of free throws made)
- p is the probability of success (the player making a free throw)
- q is the probability of failure (1-p)

In this case, n = 8 (eight free throws) and p = 0.65 (probability of making a free throw). Therefore, q = 1 - p = 1 - 0.65 = 0.35.

First, let's calculate the probability of making exactly 7 free throws and exactly 8 free throws. Then we will add them together to get the probability of making at least 7 free throws.

P(7) = (8C7) * 0.65^7 * 0.35^(8-7)
P(8) = (8C8) * 0.65^8 * 0.35^(8-8)

Using the binomial coefficient:

(nCx) = n! / (x!(n-x)!)

We have:
(8C7) = 8! / (7!(8-7)!) = 8
(8C8) = 8! / (8!(8-8)!) = 1

Calculating the probabilities:

P(7) = 8 * 0.65^7 * 0.35^(8-7)
P(8) = 1 * 0.65^8 * 0.35^(8-8)

Now, we can add these two probabilities together to get the probability of making at least 7 free throws:

P(at least 7) = P(7) + P(8)

To find the probability that the basketball player will make at least 7 of her next 8 free throws, we need to calculate the cumulative probability of making 7, 8 free throws, or 8 out of 8.

To do this, we can use the binomial probability formula:

P(x) = C(n, x) * p^x * (1 - p)^(n - x)

Where:
P(x) is the probability of getting exactly x successes
n is the number of trials or attempts
x is the number of desired successes
p is the probability of success for each trial
C(n, x) is the combination formula nCr, representing the number of possible ways to choose x successes out of n trials.

Let's apply this formula to our problem:

n = 8 (number of trials)
x = 7, 8 (number of desired successes)

First, let's calculate the probability of making exactly 7 free throws out of 8:

P(7) = C(8, 7) * (0.65)^7 * (1 - 0.65)^(8 - 7)

Using the combination formula:
C(8, 7) = 8! / (7! * (8-7)!) = 8

Substituting the values into the formula:
P(7) = 8 * (0.65)^7 * (1 - 0.65)^(8 - 7)

Next, let's calculate the probability of making exactly 8 free throws out of 8:

P(8) = C(8, 8) * (0.65)^8 * (1 - 0.65)^(8 - 8)

Using the combination formula:
C(8, 8) = 8! / (8! * (8-8)!) = 1

Substituting the values into the formula:
P(8) = 1 * (0.65)^8 * (1 - 0.65)^(8 - 8)

Finally, to find the probability of making at least 7 out of 8 free throws, we sum the individual probabilities:

P(at least 7) = P(7) + P(8)

Now you can substitute the values into the equations and calculate the final probability.