I have a statistics math problem. if someone could answer. "A basketball player makes 70% of free throws. he throws 15 free shots. what is the probability that the basketball player makes 7 or more free shots inside basket."
The requirements for an experiment's to be considered binomial are:
1. probability of success remains constant throughout the experiment
2. all trials are independent and random
3. the number of trial is constant and known.
This problem satisfies the requirements for the binomial distribution, which gives the probability of r successes out of n trials each with a probability of p as:
P(r)=C(n,r)p^r(1-p)^(n-r).
where for the present case,
C(n,r)=combination of r object out of n
=n!(r!(n-r)!)
p=0.7
n=15
r=7,8,9,...15
so
P(X>=7)=ΣP(r), r=7 to 15
=P(X=7)+P(X=8)+...P(X=15)
=0.03477+0.08113+0.14724+0.20613+0.21862+0.17004+...+0.00475
=0.98476
To solve this problem, we need to use the binomial distribution formula. The binomial distribution is used when we have a fixed number of independent trials with only two possible outcomes: success or failure.
In this case, each free throw can be considered a trial, with two outcomes: making the shot or missing it. The probability of making a shot is 70%, and the probability of missing it is 30%.
The formula for the probability of getting exactly 'k' successes in 'n' trials is:
P(X = k) = (n C k) * p^k * (1-p)^(n-k)
Where:
- (n C k) represents the combination, calculated as n! / (k! * (n-k)!)
- 'p' is the probability of success for each trial
- 'n' is the total number of trials
- 'k' is the number of successes we are interested in
Now, let's calculate the probability of making 7 or more free shots:
P(X ≥ 7) = P(X = 7) + P(X = 8) + P(X = 9) + ... + P(X = 15)
We can calculate each term separately using the formula mentioned above and sum them up.
P(X = 7) = (15 C 7) * (0.7^7) * (1-0.7)^(15-7)
P(X = 8) = (15 C 8) * (0.7^8) * (1-0.7)^(15-8)
P(X = 9) = (15 C 9) * (0.7^9) * (1-0.7)^(15-9)
...
P(X = 15) = (15 C 15) * (0.7^15) * (1-0.7)^(15-15)
Finally, add all these individual probabilities together to find the total probability of making 7 or more free shots.