Valid discrete probability distribution examples | Random variables | AP Statistics | Khan Academy

Anthony Denoon is analyzing his basketball statistics. The following table shows a probability model for the result from his next two free-throws, and so it has various outcomes of those two free-throws and then the corresponding probability: missing both free-throws 0.2, making exactly one free-throw 0.5, and making both free-throws 0.1. Is this a valid probability model?

Pause this video and see if you can make a conclusion there.

So let's talk about what makes a valid probability model.

1. The sum of the probabilities of all the scenarios needs to add up to 100%. So we would definitely want to check that.

Also, they would all have to be positive values, or I guess I should say none of them can be negative values. You could have a scenario that has a 0% probability, and so all of these look like positive probabilities. So we meet that second test that all the probabilities are non-negative.

But do they add up to 100%? So if I add 0.2 to 0.5, that is 0.7, plus 0.1, they add up to 0.8, or they add up to 80%. So this is not a valid probability model.

In order for it to be valid, all the various scenarios need to add up exactly to 100%. In this case, we only add up to 80%. If this had added up to 1.1 or 110%, then we would also have a problem. But we can just write no.

Let's do another example.

So here we are told, "You are a space alien. You visit Planet Earth and abduct 97 chickens, 47 cows, and 77 humans. Then you randomly select one Earth creature from your sample to experiment on. Each creature has an equal probability of getting selected. Create a probability model to show how likely you are to select each type of Earth creature. Input your answers as fractions or as decimals rounded to the nearest hundred."

So in the last example, we wanted to see whether the probability model was valid, was legitimate. Here, we want to construct a legitimate probability model.

Well, how would we do that? Well, the estimated probability of getting a chicken is going to be the fraction that you're sampling from. That is, our chickens, because any one of the animals is equally likely to be selected.

There are 97 of the 97 plus 47 plus 77 animals that are chickens. So what is this going to be? This is going to be 97 over 97, plus 47, plus 77. You add them up: three sevens is 21.

Then let's see: 2 plus 9 is 11, plus 4 is 15, plus 7 is 22, so 221. So 97 of the 221 animals are chickens.

I'll just write 97 over 221. They say that we can answer as fractions, so I'm just going to go that way.

What about cows? Well, 47 of the 221 are cows, so there’s a 47 over 221 probability of getting a cow.

And then last but not least, you have 77 of the 221 are humans.

Is this a legitimate probability distribution? We'll add these up. If you add these three fractions up, the denominator is going to be 221, and we already know that 97 plus 47 plus 77 is 221.

So it definitely adds up to 1, and none of these are negative, so this is a legitimate probability distribution.