Discrete random variables
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Valid discrete probability distribution examples
- [Instructor] Anthony DeNoon is analyzing his basketball statistics. The following table shows a probability model for the results from his next two free throws. And so he 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 think about what makes a valid probability model. One, the sum of the probabilities of all the scenarios need to add up to 100%. So we would definitely want to check that. And 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 .2 to .5, that is .7, plus .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, they would all, all the various scenarios need to add up exactly to 100%. In this case, we only add up to 80%. If we add it up to 1.1 or 110%, then we would also have a problem. 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 hundredth. So in the last example, we wanted to see whether the probability model was valid, was legitimate. Here, we wanna construct a legitimate probability model. Well, how would we do that? Well, the estimated probability of getting a chicken is gonna be the fraction that you're sampling from that is our chickens because any one of the animals are equally likely to be selected. 97 of the 97 plus 47 plus 77 animals are chickens. And so what is this going to be? This is gonna be 97 over. 97, 47, and 77, you add 'em up. Three sevens is a 21. And then let's see, two plus nine is 11, plus four is 15, plus seven is 22, so 221. So 97 of the 221 animals are chickens. And so I'll just write 97, 221s. They say that we can answer as fractions, so I'm just gonna go that way. What about cows? Well, 47 of the 221 are cows, so there's a 47, 221st probability of getting a cow. And then last but not least, you have 77 of the 221s are human. Is this a legitimate probability distribution? We'll add these up. If you add these three fractions up, the denominator's gonna be 221 and we already know that 97 plus 47 plus 77 is 221. So it definitely adds up to one, and none of these are negative, so this is a legitimate probability distribution.