Probability Distributions

STAT 20: Introduction to Probability and Statistics

Warmup

10:00

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For the following set of questions, you’ll be presented with a scenario and then asked to identify whether the scenario describes a

1. Binomial distribution (if so, what are \(n\) and \(p\)?)

2. Hypergeometric distribution (if so , what are \(N\), \(G\), and \(n\)?)

3. Neither of these (why)?

Binomial, hypergeometric, or neither?

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

Binomial with \(n = 20\) and \(p = \dfrac{3}{10}\)

Should probably spend a bit of time explaining this.

Binomial, hypergeometric, or neither?

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

Hypergeometric(\(N = 335\) mill, \(n = 1500\), \(G\)).

\(G\) is the number of people in the US who have read at least one book.

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

Binomial, hypergeometric, or neither?

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

• Binomial, with \(n = 1, p = \dfrac{5}{36}\)
• Or Bernoulli with \(p =\dfrac{5}{36}\).

Binomial, hypergeometric, or neither?

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 all the fruit left in the bag are equally likely on each draw). I put the fruit away after I take it out (and don’t put it back into the bag). I count the number of draws until and including the first time I draw a apple.

Neither.

Neither. It would be good for them if you were to write out all the probabilities, and explicitly point out that they are using the multiplication rule and a conditional probability.

Match the histogram to its description

Consider the following box of tickets. Match each of the following (lettered) plot to its accurate description as given on page 2 your warm up sheet:

Match the description number below to the letter of the plot on the slide:

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.

Worksheet: Probability Distributions

20:00

Break

05:00

Lab 6

25:00