Introducing Probability

STAT 20: Introduction to Probability and Statistics

Venn diagram from xkcd

Agenda

• Concept Questions
• Activity: coin tosses
• Handout

Concept Question 1

01:00

Suppose Ali and Bettina are playing a game, in which Ali tosses a fair coin \(n\) times, and Bettina wins one dollar from Ali if the proportion of heads is less than 0.4. Ali lets Bettina decide if \(n\) is 10 or 100.

Which \(n\) should Bettina choose?

This ties into the plot of proportions of heads as the number of coin tosses increases. Hopefully Bettina realizes she has a better chance of getting the number of heads far away from 0.5 with fewer tosses.

Concept Question 2

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.

B is the prob dsn of tickets C is emp dsn of 10 draws D is emp dsn of 100 draws A is emp dsn from different box

Concept Question 3

Part 1: Suppose we roll a die 4 times. The chance that we see six (the face with six spots) at least once is given by \(\displaystyle \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}\)

True or false?

Part 2: Suppose we roll a pair of dice 24 times. The chance that we see a pair of sixes at least once is given by \(\displaystyle 24 \times \frac{1}{36} = \frac{24}{36} = \frac{2}{3}\)

True or false?

02:00

Both parts are false since we are using the addition rule on events that are not mutually exclusive. It is important that you do not talk about what the actual probability is since that uses the multiplication rule and we have not discussed that in the notes. This is more an exercise in when not to use the addition rule. I plan to add a bit about this in Friday’s notes.

Concept Question 4

01:00

Consider the Venn diagram below, which has 20 possible outcomes in \(\Omega\), depicted by the purple dots. Suppose the dots represent equally likely outcomes. What is the probability of \(A\) or \(B\) or \(C\)? That is, what is \(P(A \cup B \cup C)\)?

Activity: Coin tossing

25:00

Handout (PS 3.2)