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

Lesson 1: Introduction to random variables and probability distributions- Constructing a probability distribution for random variable
- Valid discrete probability distribution examples
- Graph probability distributions
- Probability with discrete random variable example
- Probability with discrete random variables
- Theoretical probability distribution example: tables
- Theoretical probability distribution example: multiplication
- Develop probability distributions: Theoretical probabilities

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# Valid discrete probability distribution examples

Worked examples on identifying valid discrete probability distributions.

## Want to join the conversation?

- The second example includes the statement "Each creature has an equal probability of getting selected", yet the three answers are not equal. What gives?(18 votes)
- Although the wording is confusing, I believe it means that the "picker"/"space alien" is not more bias towards any type of creature, so it will not purposefully choose a chicken, cow, or human over any other type of creature. Or in other words, the space alien is picking its creatures
*at random*. But that doesn't mean the probability that it picks a chicken is equal to the probability that it picks a cow, because there are a lot more chickens than cows in the example. Another example that might clear things up would be: you randomly pick a red or blue marble out of a bag. The bag contains 10,000,000 red marbles, and 1 blue marble. What is the probability of picking a blue marble?(30 votes)

- It seems that “Valid discrete probability distribution examples” video should be before “Practice: Probability models”, not after.(15 votes)
- Just because you are given a probability problem, does not mean that you could actually solve it. Ensuring it is a valid probability model proves that:

There is a total 100% chance of anything happening at all

Prove whether it is dependent or independent

Find sampling errors. for instance:

if there is a 30% chance of selecting a green marble, 40% chance of selecting a blue marble, and 40% chance of selecting a yellow marble we know this is impossible, or that information is being withheld(1 vote)

- in the first example, it says that all scenarios must be equal to 100% (and positive). How was the second example's answer, 221, equal to a hundred percent?(2 votes)
- The total number of earth creatures is 221. 97 Chickens, 47 Cows, 77 Humans. The estimated probability is just the fraction of each type over the total amount. So, if 97+47+77=221 then, (97/221)+(47/221)+(77/221) = 221/221 = 1 or 100%. Equals 100% and are all positive values.(7 votes)

- This should come before probability models (practice).(2 votes)
- why can't we have a 0% probability?

0% means there is no chance of that event happening right?(1 vote)- We can have a 0% probability, but in most cases, this information is useless, so we neglect it.(2 votes)

- in the first example it is necessary too to check if all possibilities are displayed. if not all possibilities displayed so it is reasonable that the sum of the percentages of the displayed probabilities dont add up a whole one or 100%.

would anyone argue this!(1 vote)- Yes, in the first example, it is indeed necessary to consider whether all possible outcomes are displayed. A valid probability model must account for all possible outcomes of the experiment. If not all possibilities are displayed, it's reasonable that the sum of the probabilities won't add up to 1 (or 100%). This omission could be why the total probability in the example does not reach 100%, suggesting an incomplete model.(1 vote)

- "Each creature has an equal probability of getting selected", can there be an unequal probability of getting selected here? ex: alien is 20% more likely to pick a cow (biased reason), how will this change the answer?(1 vote)
- If the alien has a bias towards selecting a specific creature, say a 20% higher likelihood of picking a cow, this changes the probability distribution to reflect that bias. The probability model would then need to be adjusted to account for this unequal probability of selection. For example, if originally each creature had an equal chance of being selected, adjusting for a 20% higher likelihood for cows would mean recalculating the probabilities to reflect this preference, potentially by increasing the probability of selecting a cow and proportionally decreasing the probabilities for chickens and humans to ensure the total still adds up to 100%. This adjustment would make the model more complex and would require additional information on how the bias quantitatively affects the selection odds.(1 vote)

- What happens when a probability model has total outcomes greater than 100%?(1 vote)
- If a probability model has total outcomes greater than 100%, it indicates an error in the model. Probabilities must always sum up to 1 (or 100%), as this represents the total certainty of all possible outcomes. A sum greater than 100% suggests overlapping outcomes, double counting, or a misunderstanding of the probabilities, making the model invalid.(1 vote)

- what is the random variable in the alien case?(1 vote)
- The random variable in the alien abduction case is the type of Earth creature selected by the alien. It could take on one of three values: chicken, cow, or human, based on which type of creature is randomly chosen for experimentation.(1 vote)

## Video transcript

- [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.