Math / Data & Statistics

QuestionA free throw shooter has an average of making $68 \%$ of his free throws. If he throws 50 practice free throws. (HINT: use the binomlaldist function in the calculator) What is the probability that he will make between 25 and 40 of the shots? type your answer.
What is the probability that he will make at least 40 shots? type your answer_
What is the probability that he will make 30 of the shots? type your answer-

Studdy Solution

STEP 1

What is this asking? If a basketball player shoots 50 free throws, and usually makes 68% of them, what's the chance they'll make between 25 and 40, at least 40, and exactly 30? Watch out! "Between" can be tricky!
Does it include the endpoints or not?
Also, "at least" means that number or more!

STEP 2

1. Set up the problem
2. Calculate the probability of making between 25 and 40 shots
3. Calculate the probability of making at least 40 shots
4. Calculate the probability of making exactly 30 shots

STEP 3

We're dealing with a binomial distribution here!
We have a fixed number of trials ( $n = 50$ ) and the probability of success (making a free throw) is constant ( $p = 0.68$ ).
Each shot is independent of the others.

STEP 4

Our goal is to find the probability of making a certain number of shots within a given range.
We'll use the binomial probability formula for this.
Remember, it's $P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$ , where $n$ is the number of trials, $k$ is the number of successes, and $p$ is the probability of success on a single trial.

STEP 5

"Between 25 and 40" includes both 25 and 40.
So, we want $P(25 \le X \le 40)$ .
This means we need to add up the probabilities from $X=25$ all the way to $X=40$ .

STEP 6

That's a lot to calculate individually!
We can use the complement rule to make it easier.
The complement of this event is making less than 25 shots or more than 40 shots.
So we have $P(25 \le X \le 40) = 1 - [P(X < 25) + P(X > 40)]$ .

STEP 7

Using a calculator or software, we find that $P(X < 25) \approx 0.0023$ and $P(X > 40) \approx 0.0172$ .

STEP 8

Therefore, $P(25 \le X \le 40) \approx 1 - (0.0023 + 0.0172) = 1 - 0.0195 = \textbf{0.9805}$ .
There's a very high chance they'll make between 25 and 40 shots!

STEP 9

"At least 40" means 40 or more.
So we want $P(X \ge 40)$ .
This is equal to $P(X=40) + P(X=41) + ... + P(X=50)$ .

STEP 10

Again, using a calculator or software, we find that $P(X \ge 40) \approx \textbf{0.0172}$ .

STEP 11

This is a straightforward calculation!
We just need to find $P(X=30)$ .

STEP 12

Using the binomial probability formula (or a calculator), $P(X=30) \approx \textbf{0.0261}$ .

STEP 13

The probability of making between 25 and 40 shots is approximately 0.9805. The probability of making at least 40 shots is approximately 0.0172. The probability of making exactly 30 shots is approximately 0.0261.

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Studdy Solution

STEP 1

What is this asking? If a basketball player shoots 50 free throws, and usually makes 68% of them, what's the chance they'll make between 25 and 40, at least 40, and exactly 30? Watch out! "Between" can be tricky!
Does it include the endpoints or not?
Also, "at least" means that number or more!

STEP 2

1. Set up the problem
2. Calculate the probability of making between 25 and 40 shots
3. Calculate the probability of making at least 40 shots
4. Calculate the probability of making exactly 30 shots

STEP 3

We're dealing with a binomial distribution here!
We have a fixed number of trials ( $n = 50$ ) and the probability of success (making a free throw) is constant ( $p = 0.68$ ).
Each shot is independent of the others.

STEP 4

STEP 5

"Between 25 and 40" includes both 25 and 40.
So, we want $P(25 \le X \le 40)$ .
This means we need to add up the probabilities from $X=25$ all the way to $X=40$ .

STEP 6

That's a lot to calculate individually!
We can use the complement rule to make it easier.
The complement of this event is making less than 25 shots or more than 40 shots.
So we have $P(25 \le X \le 40) = 1 - [P(X < 25) + P(X > 40)]$ .

STEP 7

Using a calculator or software, we find that $P(X < 25) \approx 0.0023$ and $P(X > 40) \approx 0.0172$ .

STEP 8

Therefore, $P(25 \le X \le 40) \approx 1 - (0.0023 + 0.0172) = 1 - 0.0195 = \textbf{0.9805}$ .
There's a very high chance they'll make between 25 and 40 shots!

STEP 9

"At least 40" means 40 or more.
So we want $P(X \ge 40)$ .
This is equal to $P(X=40) + P(X=41) + ... + P(X=50)$ .

STEP 10

Again, using a calculator or software, we find that $P(X \ge 40) \approx \textbf{0.0172}$ .

STEP 11

This is a straightforward calculation!
We just need to find $P(X=30)$ .

STEP 12

Using the binomial probability formula (or a calculator), $P(X=30) \approx \textbf{0.0261}$ .

STEP 13

The probability of making between 25 and 40 shots is approximately 0.9805. The probability of making at least 40 shots is approximately 0.0172. The probability of making exactly 30 shots is approximately 0.0261.

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Studdy Solution

STEP 1

What is this asking? If a basketball player shoots 50 free throws, and usually makes 68% of them, what's the chance they'll make between 25 and 40, at least 40, and exactly 30? Watch out! "Between" can be tricky!Does it include the endpoints or not?Also, "at least" means that number or more!

STEP 2

1. Set up the problem2. Calculate the probability of making between 25 and 40 shots3. Calculate the probability of making at least 40 shots4. Calculate the probability of making exactly 30 shots

STEP 3

We're dealing with a **binomial distribution** here!We have a fixed number of trials (n=50n = 50n=50) and the probability of success (making a free throw) is constant (p=0.68p = 0.68p=0.68).Each shot is **independent** of the others.

STEP 4

STEP 5

"Between 25 and 40" **includes** both 25 and 40.So, we want P(25≤X≤40)P(25 \le X \le 40)P(25≤X≤40).This means we need to add up the probabilities from X=25X=25X=25 all the way to X=40X=40X=40.

STEP 6

STEP 7

Using a calculator or software, we find that P(X<25)≈0.0023P(X < 25) \approx 0.0023P(X<25)≈0.0023 and P(X>40)≈0.0172P(X > 40) \approx 0.0172P(X>40)≈0.0172.

STEP 8

Therefore, P(25≤X≤40)≈1−(0.0023+0.0172)=1−0.0195=0.9805P(25 \le X \le 40) \approx 1 - (0.0023 + 0.0172) = 1 - 0.0195 = \textbf{0.9805}P(25≤X≤40)≈1−(0.0023+0.0172)=1−0.0195=0.9805.There's a very high chance they'll make between 25 and 40 shots!

STEP 9

"At least 40" means 40 or more.So we want P(X≥40)P(X \ge 40)P(X≥40).This is equal to P(X=40)+P(X=41)+...+P(X=50)P(X=40) + P(X=41) + ... + P(X=50)P(X=40)+P(X=41)+...+P(X=50).

STEP 10

Again, using a calculator or software, we find that P(X≥40)≈0.0172P(X \ge 40) \approx \textbf{0.0172}P(X≥40)≈0.0172.

STEP 11

This is a straightforward calculation!We just need to find P(X=30)P(X=30)P(X=30).

STEP 12

Using the binomial probability formula (or a calculator), P(X=30)≈0.0261P(X=30) \approx \textbf{0.0261}P(X=30)≈0.0261.

STEP 13

The probability of making between 25 and 40 shots is approximately 0.9805. The probability of making at least 40 shots is approximately 0.0172. The probability of making exactly 30 shots is approximately 0.0261.

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Studdy solves anything!

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? If a basketball player shoots 50 free throws, and usually makes 68% of them, what's the chance they'll make between 25 and 40, at least 40, and exactly 30? Watch out! "Between" can be tricky!
Does it include the endpoints or not?
Also, "at least" means that number or more!

1. Set up the problem
2. Calculate the probability of making between 25 and 40 shots
3. Calculate the probability of making at least 40 shots
4. Calculate the probability of making exactly 30 shots

We're dealing with a binomial distribution here!
We have a fixed number of trials ( $n = 50$ ) and the probability of success (making a free throw) is constant ( $p = 0.68$ ).
Each shot is independent of the others.

"Between 25 and 40" includes both 25 and 40.
So, we want $P(25 \le X \le 40)$ .
This means we need to add up the probabilities from $X=25$ all the way to $X=40$ .

Using a calculator or software, we find that $P(X < 25) \approx 0.0023$ and $P(X > 40) \approx 0.0172$ .

Therefore, $P(25 \le X \le 40) \approx 1 - (0.0023 + 0.0172) = 1 - 0.0195 = \textbf{0.9805}$ .
There's a very high chance they'll make between 25 and 40 shots!

"At least 40" means 40 or more.
So we want $P(X \ge 40)$ .
This is equal to $P(X=40) + P(X=41) + ... + P(X=50)$ .

Again, using a calculator or software, we find that $P(X \ge 40) \approx \textbf{0.0172}$ .

This is a straightforward calculation!
We just need to find $P(X=30)$ .

Using the binomial probability formula (or a calculator), $P(X=30) \approx \textbf{0.0261}$ .