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STAT 20: Introduction to Probability and Statistics

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

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

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

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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?

Linda is a bank teller.

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

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

Recall the following picture from the notes:

Let’s simulate tossing a coin ten times:

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>
1 Heads 15
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>
1 Heads 251
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.

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 \(1000\).

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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?

`02:00`

Consider the box of tickets shown below.

The plots below show:

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

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

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?

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`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 \(P(A \cup B \cup C)\)?

`25:00`