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### Course: AP®︎/College Statistics > Unit 8

Lesson 5: Introduction to the binomial distribution- Binomial variables
- Recognizing binomial variables
- 10% Rule of assuming "independence" between trials
- Identifying binomial variables
- Binomial probability example
- Generalizing k scores in n attempts
- Free throw binomial probability distribution
- Graphing basketball binomial distribution
- Binompdf and binomcdf functions
- Binomial probability (basic)
- Binomial probability formula
- Calculating binomial probability

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# Graphing basketball binomial distribution

Sal graphs the results of using the binomial distribution to find the probabilities of making different numbers of free throws.

## Want to join the conversation?

- In the 'Binomial distribution' video, the probability was calculated by finding the total number of events and then using the combinatorics formula to find the chance of X occurring however many times and dividing that by the total number of possibilities to get the probability. On the other hand in the 'Probability of making 2 shots in 6 attempts' the value derived from using combinatorics is multiplied with the probabilities to get the final answer. Why is there a difference in the two approaches ? Sorry my question is a little confusing(6 votes)
- He just skipped the multiplication part since both outcomes are equaly likely. If you multiply by the probabilities raised to the correct exponant, like he does here, it comes to the same result.(2 votes)

- Is it true that even binomial distributions that don't have 50-50 probability of failure-success approaches normal distribution as the number of trials goes to infinity? And if so, do binomial distributions that don't have 50-50 approaches normal slower than the 50-50 ones?(4 votes)
- Yes, according to the Central Limit Theorem and the De Moivre-Laplace theorem, binomial distributions tend to approach a normal distribution as the number of trials n increases, even if the probability of success p is not 0.5. However, the speed at which they approach a normal distribution can indeed vary. Distributions with p closer to 0.5 tend to "normalize" quicker than those with p skewed heavily towards 0 or 1. This is because the variance is maximized at p = 0.5, making the distribution's shape more symmetric and thus more closely approximating a normal distribution even with a lower number of trials.(2 votes)

- at0:08, you said "size". Did you mean to?(3 votes)
- At0:08, Sal said "6 free throws". The caption is incorrect it says "size".(4 votes)

- when you did the calculation for P(X=i) why didnt you show that you mutipy by the "6 choose i" , when u just mutipy the way you did you dont get the correct percentages(2 votes)
- When calculating P(X = i) for a binomial distribution, the correct approach indeed involves multiplying by (6Ci), where i represents the number of successful outcomes (made free throws). The formula for the probability of exactly i successes in n trials is P(X = i) = (nCi)p^i(1 − p)^n−i, where (nCi) is the binomial coefficient representing the number of ways to choose i successes out of n attempts, p is the probability of success on a single trial, and (1 − p) is the probability of failure. If the multiplication by (6Ci) wasn't explicitly shown or mentioned in the calculation process, it's a crucial step that must be included to obtain the correct probabilities.(1 vote)

- This a histogram ? I thought histograms requires ranges.(2 votes)
- A histogram is a type of bar chart that represents the frequency of distinct outcomes or ranges of outcomes in a dataset. While histograms typically display ranges of values on the x-axis (especially for continuous data), they can also represent discrete data categories or specific numerical outcomes, such as the number of successes in a binomial distribution. In the context of binomial distributions, each bar represents the probability of achieving exactly i successes (not a range), making the chart a discrete probability distribution plot. While it shares similarities with histograms, such as using bars to display quantities, it's more accurately described as a probability distribution plot for discrete outcomes.(1 vote)

- Hi, what are the formulas to calculate the coefficients Sal refers to (the uniform ones)?

Thanks!(2 votes)- The coefficients Sal refers to are binomial coefficients, which are calculated using the formula: (nCk) = n! / k!(n−k)!, where n! (n factorial) is the product of all positive integers up to n, and k is the number of successes. These coefficients represent the number of distinct ways k successes can occur out of n trials and are a key part of the binomial probability formula.(1 vote)

## Video transcript

- In the last video, we worked through essentially the probability distribution for this random variable defined as the number of free throws you make when taking size free throws, assuming you have a 70
percent free throw percentage, and I suggested, hey, why
don't you visualize this? Draw the graph of this
probability distribution, this binomial probability distribution. And when I thought about it, I said, well, I too would enjoy graphing it, and we might as well do it together, because whenever you graph these things it makes it very visual, and kind of the shape of a
binomial distribution like this. So let's do that. So let me maybe move over
to the right a little bit, I really just need to
be able to keep track of these things right over here. Let me draw some lines. If I were to just draw one line there, and then another line here, and then we have the
different percentages. So let's do that. See, the highest one is
a little over 32 percent. So maybe we'll go as
high as 40 percent here. 40 percent, and then
this would be 20 percent. 20, that looks about half way, 20 percent. This would be 10 percent. And this would be 30 percent. And then in this axis, let's
do the different values that the random variable could take on. So the random variable taking on the value zero. The random variable
taking on the value one. Zero, one, the random
variable taking on two. Two, we're almost there. Let's see, three, and then four. Four, and then five. Five, and then finally six. Take x equals six. And then six, and now let's
just graph all of these. So this first one, zero point one percent, well, that's barely going to register on this graph right here, so I'll just kind of
just give it a little bit of a showing right over there. Showing (mumbles) due to that green color. So, let me make sure. So, in that green color, you're going to have just
a little bit of a showing. One, as well, is kind of barely a showing. So it shows up a little bit more. So let me draw it like that. That is one percent, right over there. Now, two is six percent,
which on this scale, is going to be about that high. So let me draw it like that. So that is two. So that is six percent right there. X equaling three, 18
and a half percent shot of that happening... So 18 and a half, it's a hand drawn chart, or histogram, so you have to bear with me. So it's roughly there, and then four was 32.4 percent, so that is up here. So 32.4 percent looks like that. Let me shade that in, 32.4 percent. And then five was 30.3 percent, 30.3 percent, slightly
lower, just like that. And it will look like this. 30.3 percent, and finally,
six is 11.8 percent. So really, this whole video was just an exercise in making a histogram, but it's useful, because
to actually visualize what the distribution looks like. And what's really interesting, is to think about how does this change as you change the free throw percentage, or as you change the
number of shots you take, how does this change this
binomial distribution? And you can do that on a spreadsheet, and actually see how that all works out.