If a basketball player has a shooting percentage of 60% , find the probability that the player will make at least 4 of her next 6 shots?
"at least 4" = 4, 5, or 6.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
Online, “*” is used to indicate multiplication to avoid confusion with “x” as an unknown.
Online "^" is used to indicate an exponent, e.g., x^2 = x squared.
P(4) = .6^4 * .4^2
P(5) = .6^5 * .4
P(6) = .6^6
Either-or probabilities are found by adding the individual probabilities.
P(4,5 or 6) = P(4) + P(5) + P(6)
Or you could explore combinations and permutations.
https://www.google.com/search?client=safari&rls=en&q=combinations+permutations+probability&ie=UTF-8&oe=UTF-8&gws_rd=ssl
To find the probability that the player will make at least 4 of her next 6 shots, we can use the binomial probability formula.
The formula for the 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 number of combinations of n items taken k at a time.
p is the probability of success for each trial.
n is the number of trials.
In this case, the player has a shooting percentage of 60% or 0.6 as the probability of success. The player will take 6 shots, so n = 6. We want to find the probability of making at least 4 shots, which means k >= 4.
To find the probability of making *exactly* k shots, we need to sum the probabilities of making 4, 5, and 6 shots:
P(X=4) + P(X=5) + P(X=6)
Now, let's calculate the individual probabilities:
P(X=4) = C(6,4) * 0.6^4 * (1-0.6)^(6-4)
P(X=5) = C(6,5) * 0.6^5 * (1-0.6)^(6-5)
P(X=6) = C(6,6) * 0.6^6 * (1-0.6)^(6-6)
Calculating these probabilities:
P(X=4) = 15 * 0.6^4 * 0.4^2 = 0.27648
P(X=5) = 6 * 0.6^5 * 0.4^1 = 0.2304
P(X=6) = 1 * 0.6^6 * 0.4^0 = 0.046656
Now, we can sum these probabilities:
P(X>=4) = P(X=4) + P(X=5) + P(X=6)
= 0.27648 + 0.2304 + 0.046656
= 0.553536
Therefore, the probability that the player will make at least 4 of her next 6 shots is approximately 0.553536 or 55.4%.
To find the probability that the player will make at least 4 of her next 6 shots, we can use the binomial probability formula. The formula is given by:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of exactly k successes
- C(n, k) is the number of combinations of n items taken k at a time
- p is the probability of success on a single attempt
- (1-p) is the probability of failure on a single attempt
- n is the total number of attempts
In this case, we want to find the probability of making at least 4 out of 6 shots. This means we need to calculate the individual probabilities for making exactly 4, 5, and 6 shots, and then sum them up.
Step 1: Calculate the probability of making exactly k shots:
- Probability of making a single shot (p) = 0.6 (60%)
- Number of attempts (n) = 6
Step 2: Calculate the probability of making exactly 4 shots:
- k = 4
- P(X = 4) = C(6, 4) * 0.6^4 * (1-0.6)^(6-4)
Step 3: Calculate the probability of making exactly 5 shots:
- k = 5
- P(X = 5) = C(6, 5) * 0.6^5 * (1-0.6)^(6-5)
Step 4: Calculate the probability of making exactly 6 shots:
- k = 6
- P(X = 6) = C(6, 6) * 0.6^6 * (1-0.6)^(6-6)
Step 5: Sum up the probabilities of making at least 4 shots:
- P(X >= 4) = P(X = 4) + P(X = 5) + P(X = 6)
By calculating these probabilities using the binomial formula, you can find the probability that the player will make at least 4 of her next 6 shots.