Probability Distributions

STAT 20: Introduction to Probability and Statistics

Agenda

• Announcements
• Probability refresher (last week)
• Break
• Concept questions: Probability Distributions
• PS 8: Probability Distributions

Announcements

• No lab today

• Problem Set 8 (paper max. 3) due next Tuesday at 9am

• RQ: Random Variables due Wednesday 11:59pm

You may need to review the concepts as I said in slack, depending on how they do in the pe refresher

Probability Refresher

25:00

Concept Questions

02:00

Consider the box of tickets shown below.

The plots below show:

1. The probability histogram for the value of a ticket drawn at random from the box
2. An empirical histogram for which the data were generated by drawing 10 tickets from the box with replacement
3. An empirical histogram for which the data were generated by drawing 100 tickets from the box with replacement
4. An empirical histogram generated by 20 draws from a different box.

Identify which is which by matching the letters to the numbers.

Since you have to scroll, maybe draw the box on the board. Scroll to see graphs

• B-1 the prob dsn of tickets
• D-2 emp dsn of 10 draws
• C-3 emp dsn of 100 draws
• A-4 emp dsn from different box

Binomial, hypergeometric, or neither?

01:00

Does the quantity described below have a probability distribution that is binomial (if so, what are \(n\) and \(p\)?), hypergeometric (if so , what are \(N\), \(G\), and \(n\)?), or neither (why)?

Roll a fair ten-sided die 20 times. We count the number of times we roll a multiple of 3.

Binomial, hypergeometric, or neither?

Binomial(20, 3/10). The pollev will have 4 choices, and then figuring out the parameters can be a classroom discussion. The fourth choice is “Need more info”

01:00

Does the quantity described below have a probability distribution that is binomial (if so, what are \(n\) and \(p\)?), hypergeometric (if so , what are \(N\), \(G\), and \(n\)?), or neither (why)?

YouGov surveyed 1,500 adult US citizens last December and counted the number of respondents who had read at least one book in 2023. The population of the US is about 335 million.

Binomial, hypergeometric, or neither?

Hypergeometric(\(N = 335\) mill, \(n = 1500\), G), where G is not known, being the number of people who have read at least one book . Can remark that we could use the binomial to model this rather than HG, and that we will discuss sample surveys in this unit. Disturbingly, only 54% of respondents had read at least one book. Here is the link: https://today.yougov.com/entertainment/articles/48239-54-percent-of-americans-read-a-book-this-year

01:00

Are we describing a probability distribution below that is binomial? Or is it hypergeometric? Or do neither apply?

A six-sided die is tossed twice, and we check if the sum of the spots is 8 or not.

Binomial, hypergeometric, or neither?

Binomial, with \(n = 1, p = 5/36\) or Bernoulli with \(p = 5/36\).

01:00

Are we describing a probability distribution below that is binomial? Or is it hypergeometric? Or do neither apply?

A bag that has 6 pieces of fruit: 2 mangoes, 3 apples, and 1 orange. I reach into the bag and draw out one fruit at a time, selecting each fruit at random (so they are equally likely on each draw). I count the number of draws until and including the first time I draw a apple.

Binomial, hypergeometric, or neither?

Neither. It would be useful at this point to write out what is going on, and point out that they are using the multiplication rule and a conditional probability.

Break

05:00

PS 8

45:00

End of lecture