# Introducing Probability

STAT 20: Introduction to Probability and Statistics

## Agenda

• Review of ideas in notes
• Concept Questions
• Activity: coin tosses
• PS 7: Introduction to chance

# Concept review: Rules

## Rules of probability

Let $\Omega$ be the outcome space, and let $P(A)$ denote the probability of the event $A$. Then we have:

1. $P(A) \ge 0$
2. $P(\Omega) = 1$
3. If $A$ and $B$ are mutually exclusive ($A \cap B = \{\}$), then $P(A \cup B) = P(A) + P(B)$

# Concept Question 1

## The Linda Problem

01:00

The Linda problem is from a very famous experiment conducted by Daniel Kahneman and Amos Tversky in 1983 (The version below is from the book Thinking, Fast and Slow by Kahneman, page 156):

Linda is thirty-one years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in antinuclear demonstrations.

Which alternative is more probable?

1. Linda is a bank teller.

2. Linda is a bank teller and is active in the feminist movement.

Kahneman, Daniel. Thinking, Fast and Slow (p. 158). Farrar, Straus and Giroux.

# Concept review: Simulation, empirical distributions

## Rolling a die

Recall the following picture from the notes:

## Coin tosses

Let’s simulate tossing a coin ten times:

set.seed(12345)

tosses <- sample(coin1, 10, replace = TRUE)

data.frame(tosses) %>%
group_by(tosses) %>%
summarise(n = n())
# A tibble: 2 × 2
tosses     n
<chr>  <int>
2 Tails      7

## Coin tosses

What about if I toss it fifty times?

set.seed(12345)

tosses <- sample(coin1, 50, replace = TRUE)

data.frame(tosses) %>%
group_by(tosses) %>%
summarise(n = n())
# A tibble: 2 × 2
tosses     n
<chr>  <int>
2 Tails     35

This doesn’t look so good… Maybe tossing five hundred times will improve the split:

set.seed(12345)

tosses <- sample(coin1, 500, replace = TRUE)

data.frame(tosses) %>%
group_by(tosses) %>%
summarise(n = n())
# A tibble: 2 × 2
tosses     n
<chr>  <int>
2 Tails    249

We see that as the number of tosses increases, the distribution of tosses begins to look closer to 50% heads and 50% tails.

## Looking at proportions:

Here is a plot of the proportion of tosses that land heads when we toss a coin $n$ times, where $n$ varies from $1$ to $1000$.

# Concept Question 2

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?

# Concept Question 3

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.

# Concept Question 4

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

# Concept Question 5

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