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Binomial Distribution
Common Core: S.MD.1

Binomial Distribution

Author: Ryan Backman
Description:

Identify criteria of a binomial distribution

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Tutorial

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Hi. This tutorial covers the binomial distribution. All right. The binomial distribution is a type of distribution that occurs when the following four conditions are met. Number one, there are a fixed number of trials with only two outcomes. Two, the trials are independent. So the outcome of the first trial has no effect on the outcome of a second trial.

Number three, the probability of the outcomes does not change. OK. So those probabilities stay the same for each trial. And number four, the variable of interest is the number of successes.

So in number one, when we're talking about only two outcomes, we're talking about either a success or a failure. So this is binomial, bi- meaning two. So we have two outcomes here.

OK. So the binomial distribution can be represented with a formula. And what this does is-- so instead of just using capital P, I'm going to use Prob, so for probability. And what k represents is the number of successes. So if we're looking for the probability of a certain number of successes, we can calculate that probability using this formula.

So k is the number of successes. n is the number of trials. And p is the probability of success.

Notice this notation here. Sometimes we'll say that as n choose k. It counts the number of combinations of k successes in n trials. So this basically tells me how many different combinations there are of k successes among n trials. Sometimes you'll also see this as n and then a capital C and then a k. And we'll look at how to get this on the calculator.

All right. So let's take a look at an example. So in the winter, I play the sport of broomball. Here's a picture of a broomball rink. It's pretty similar to hockey. From my experience, the probability that I will score on a penalty shot is about 65%. So a penalty shot-- you just have an open shot on the goalie, and I'm about a 65% shooter there.

Suppose that next season I will take three penalty shots. What is the probability that I will score on two of the shots? OK. So what I'm trying to figure out here is, what is the probability of 2? So again, p meaning probability-- probability of 2 goals.

Now, let's make sure that we do have a binomial setting here. So let's make sure our four conditions are met. So one, a fixed number of trials with only two outcomes. Is that condition met?

I would say yes. We have only three shots. So that's going to be a fixed number of trials, so there'll be three trials. And two outcomes. Yes, there are two outcomes. Either I make the shot or miss the shot. So one is good.

Number two, the trials are independent. So the outcome of the first shot won't affect the outcome of the second shot. I would say that that would be satisfied here, that whether or not I make or miss the first won't have an effect on whether I make or miss the second.

And three, the probability of the outcomes does not change. That probability of success is always going to remain at 65%. And number three, the variable of interest is the number of successes. Yes, that is met. A success, in this case, would represent a goal.

OK. So let's apply that formula to calculate p of 2. So what we need, again, are a couple-- we need-- there are a couple different parameters here. So we know that n is 3 here because we're doing three shots. Lowercase p, which is the probability of success, is 0.65.

And now at this point, we can use the formula to calculate this probability. So it's going to be the probability of 2. And that's going to equal-- now, it was n choose k. So n is 3, and k is 2. I'm looking for two successes.

Then I need to multiply that by the probability of success to the power of k. So it's going to be 0.65 to the power of k, which is 2. Times 1 minus the probability of success-- whoops-- so 1 minus 0.65. That's going to equal 0.35, if I subtract it there.

And then to the power of n minus k, so 3 minus 2. So now if we match that up against the formula that we had here, we can see that we have n choose k here, probability of success to the power of k, so 0.65 to the power of k, and then 1 minus 0.65 to the power of n minus k.

Now, I'm going to do most of this in the calculator. I'm actually going to figure out what this is first-- the number of ways of-- number of combinations to get two successes among three trials. And the way I'm going to do that on the calculator, I'm going to go 3. Now, my calculator has a function, this NCR function-- NCR 2. And that equals 3. So there are three ways to get two successes in three trials.

So that's going to equal 3. Times-- now, it's going to be 0.65 squared-- whoops-- 0.65 to the second power. Now times 1 minus 0.65 to the power of-- now, 3 minus 2 is going to equal 1. So I'm going to do that to the power of 1 there. So that's going to be 0.147875.

OK. So now I need to multiply that by 3. And I end up with 0.443625, which is about 44%. So about 44% there is going to end up being my probability. So what that means is that I have a 44% chance that I will make 2 out of the 3 penalty shots we take here. All right. So that is how to calculate probability using a binomial distribution.

And this has been your tutorial on the binomial distribution. Thanks for watching.

Terms to Know
Binomial Distribution

The distribution of the number of successes that occur within n independent trials of a chance experiment with two outcomes per trial and p probability of success per trial.

Formulas to Know
Binomial Distribution

P left parenthesis X equals k right parenthesis space equals space open parentheses table row n row k end table close parentheses times p to the power of k times open parentheses 1 minus p close parentheses to the power of n minus k end exponent
k space i s space t h e space n u m b e r space o f space s u c c e s s
n space i s space t h e space n u m b e r space o f space t r i a l s
p space i s space t h e space p r o b a b i l i t y space o f space s u c c e s s