Probability Foundations

STAT 20: Introduction to Probability and Statistics

Warmup Questions

10:00

At the start of class you could hand out index cards and collect each students birthdays. I suggest that you start the timer on your speaker notes so that you can go back and keep an eye on it. Meanwhile have the tutors go around and collect the index cards, and start sorting them to look for matches.

If I roll a fair six-sided die, what is the chance that I will roll a multiple of 3 OR a multiple of 4?

Multiples of 3 in \(\{1, \ldots, 6\}\) are \(\{3, 6\}\), and the only multiple of 4 is \(\{4\}\). These two events don’t overlap (they are mutually exclusive), so by the addition rule, \(P(\text{multiple of 3 } OR \text{ multiple of 4}) = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}\).

We have two events \(A\) and \(B\) in an outcome space \(\Omega\). If \(P(A) = 0.7\) and \(P(B) = 0.5\), can \(A\) and \(B\) be mutually exclusive? Draw a Venn diagram and say what is the biggest the intersection of \(A\) and \(B\) can be (an upper bound on \(P(A \cap B)\)), and what is the smallest it can be.

• No. If \(A\) and \(B\) were mutually exclusive, \(P(A \cup B) = P(A) + P(B) = 1.2\).
• We know that \(P(A \cup B) \le 1\). This means that \(A\) and \(B\) must overlap.
• The intersection is largest when one of the events is wholly inside the other.
• Here, \(P(B) < P(A)\), so we must have that \(B\) sits inside \(A\) (\(B \subset A\)).
• Therefore, \(A \cap B = B\) and \(P(A \cap B) = P(B) = 0.5\).
• We saw that \(P(A) + P(B) = 1.2\), but the biggest the union can be is 1.
• This means that the sets have to overlap enough to make their union 1.
• The smallest intersection has to be the amount greater than 1, which is \(0.2\).

This might be worth working out on the board to make sure they understand. But do NOT spend too much time on it.

Do you share a birthday with someone in the room? We have roughly 80 people in the classroom. Do you think the chance that you share a birthday with someone is more than 50% (just the day, for example, February 22)? How many people would you need in the room before the chance that two people have the same birthday is more than 50%? How would you even begin to think about this chance?

This is just to get them talking and thinking about a fun problem. They will use R to work out the probability in the lab.

Concept review: Axioms

Axioms (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 \(P(A \cup B) = P(A) + P(B)\)

Concepts to review: (KEEP THIS BRIEF, OR INCORPORATE INTO CQ’s) - The first two rules of probability - unions and intersections - mutually exclusive events and the addition rule - good idea to draw Venn diagrams here - Use rule 3 to write down the complement rule, and show what A^C means

Concept Questions

Using Venn diagrams to compute probability

00:30

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

Ask them what is the quickest way to do this. Maybe even give them only 15 or 30 seconds.

Probability of two events…

xkcd comic showing two people discussing what it means to have a 50-50 chance

https://imgs.xkcd.com/comics/prediction.png

Maybe a bit of discussion here about situations in which two things can happen, but what do they think about the chance of each eg. coin toss, even/odd number on a die roll, winning the lottery vs not winning, getting an A in stat 20 vs not etc.

Box model

Suppose we toss a pair of fair six-sided dice, and sum the spots (such as when we play Monopoly). We want to model this using a box of tickets. Will the box shown here work? If so, how many times should we draw, and if not, why not?

Tossing a fair coin: how many heads do you expect?

Coin tosses: 10 times

We can simulate coin tosses and see if the simulations justify our intuitive understanding of what happens when we toss a fair coin. Go over code and review set.seed() and sample(). Maybe change arguments and see what happens.

set.seed(12345)

coin <- c("Heads", "Tails")
tosses <- sample(coin, 10, replace = TRUE)
data.frame(tosses) |>
  group_by(tosses) |> 
  summarise(n = n())

# A tibble: 2 × 2
  tosses     n
  <chr>  <int>
1 Heads      3
2 Tails      7

Coin tosses: 50 times

Comment on the fact that it is not a 50-50 split.

set.seed(12345)

tosses <- sample(coin, 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

Coin tosses: 500 times

Maybe tossing five hundred times will improve the split:

set.seed(12345)

tosses <- sample(coin, 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 split of heads and tails begins to look closer to 50-50.

Looking at the proportion of tosses that land heads:

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 \(1000\).

Be sure to explain what they are looking at. We see that at the beginning, for a small number of tosses, the proportion of times that the coin lands heads is all over the place, but it eventually settles down to be around 0.5. This verifies our intuition that if we toss a fair coin, the proportion of times that the coin lands heads should be about 0.5. This idea of the probability of a particular outcome as a long run proportion of times that we see that outcome is called the frequentist theory of probability, and we will be using this theory in our class. (A different theory of probability uses a subjective notion of probability, but we won’t get into that at this time.) We will think about the probability of heads as the long-run relative frequency, or the proportion of times the coin will land heads if we toss it many, many times. This fits with our intuition that if we have a fair coin, that means that each of the two possible outcomes will occur roughly the same number of times when we toss it over and over again. This is the justification for calling the two outcomes equally likely and allowing us to define the probability of heads to be 1/2.

How many tosses?

00:30

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.

Coin tossing game: You vs the prof!

• Make groups of about 6-8 students each.
• Each group should divide its members into two subgroups (who should not communicate)
• Prof leaves the classroom, and tutors supervise.
• In each group, one of the subgroups will flip a quarter 100 times and record the results as a sequence of 0’s and 1’s, with 1 representing the coin landing heads and 0 representing the coin landing tails.
• The other subgroup will *make up * a sequence of length 100 0’s and 1’s which is supposed to represent the result of 100 coin flips and write this on a sheet of paper - but they mustn’t actually flip a coin, or use their phone, or use their computer etc. They also mustn’t talk to the other subgroup.
• Now have someone from each of the subgroup go to the board and write their sequences. Each group therefore puts up a pair of sequences, one truly obtained by flipping coins and the other a fake one.
• When all the groups are done, call the prof. back in and have them guess, for each group, which sequence is real and which is fake.

Worksheet: Probability Foundations

20:00

Lab 5: Probability

25:00