Review of the Binomial Distribution

Definition The Binomial Distribution is the distribution of ___?___

Definition The Binomial Distribution is the distribution of COUNTS. It counts the number of successes in a certain number of trials.

COUNT the number of shots he or she made. For example, if one wanted to find out how many free throws a basketball player makes in one game, one would… COUNT the number of shots he or she made.

Wait…so does that mean the distribution of the shots made is binomial? The answer is…NO!

Why isn’t it a binomial distribution if it depends on counts? To answer that question, we must look at the Binomial Setting

The Binomial Setting Fixed number of n trials Independence Two possible outcomes: success or failure Same probability of a success for each observation If it FITS, it’s binomial.

Going back to the example, in what ways did it not satisfy the binomial setting? First of all, there is no set number of n trials. In a basketball game, one cannot predict or set how many free-throws the player is going to shoot. Second, there isn’t independence or a set probability of a success in each shot they take. The player can improve or get worse with more shots taken.

What is a good example that satisfies the binomial setting? Although there are many examples that satisfy the binomial setting, the Coin Toss experiment is the example we’re going to use.

How can you carry out this experiment so it doesn’t go against the binomial setting? You select how many times you want to toss the coin. Decide which side (heads or tails) is going to be the “success” when it lands. Make the coin “fair”, meaning that the probability of landing either heads or tails is .5. Independence is a given, unless one can toss the coin in a way that one outcome is favored over the other.

To do the experiment, you can... Toss the coins physically and record your observations Run a simulation on a calculator or on a computer, provided by your teacher.

There is a formula that can help us figure out the probabilities of getting a certain number of successes in a certain number of trials.

What is that formula? (Binomial Probability)

Can you explain this formula? (Test Question) To find the probability of k successes, you find , which is the number of sequences containing k number of successes. Then, you multiply by the probability of k successes and probability of n-k failures.

If we were to toss 10 coins, the probability of getting 6 successes is...

How would we do this on the calculator? You can go to the MATH key, then to PRB and find the function nCr. This is the same as . Put the value of n before nCr and the number of successes you want after nCr. Then you multiply by . Or you can use the function binompdf. Binompdf(# of trials, p, x)

What’s the difference between binompdf and binomcdf? If we were looking for the probability of getting 6 heads out of 10 tosses, then binompdf only finds the likelihood of getting 6 successes. Binomcdf adds up all the probability of successes up to that certain number, 6 in this case, of successes, starting from 0 to k.

What is the mean and the standard deviation of the binomial distribution?

Is this distribution normal? No, because it depends on counts and counts are related to proportions. Proportions are NEVER normal.

You use Normal Approximation for binomial distributions. What happens when n gets so large that it becomes awkward to use the formula? You use Normal Approximation for binomial distributions.

When can we use Normal Approximation? When np is greater than or equal to 10 and n(1-p) is greater than or equal to 10. Once this requirement is met, you can treat it like a normal distribution by using normcdf on your calculator. But when you use normal approximation, the probability you get is an approximation, while the probability you get through the formula is exact.

Summary Binomial distribution depends on counts (never normal). FITS: the binomial setting The formula: Parameters: Probability obtained through formula: exact Probability obtained through approximation: not exact