# Computing Chances

STAT 20: Introduction to Probability and Statistics

## Agenda

• PS 7
• Concept Review
• Concept Questions
• Break
• PS 8 (computing probabilities)
• Break

# Concept questions & review

## Rules

• Multiplication rule

For two events $A$ and $B$, $P(A \text{ and } B) = P(A \vert B) P(B)$

• Complement rule

$P(A^C) = 1 - P(A)$

01:00

Flip 3 coins, one at a time. Define the following events:

$A$ is the event that the first coin flipped shows a head

$B$ is the event that the first two coins flipped both show heads

$C$ is the event that the last two coins flipped both show tails

The events A and B are: ________

01:00

Flip 3 coins, one at a time. Define the following events:

$A$ is the event that the first coin flipped shows a head

$B$ is the event that the first two coins flipped both show heads

$C$ is the event that the last two coins flipped both show tails

The events $A$ and $C$ are: ________

Suppose we draw 2 tickets at random without replacement from a box with tickets marked {1, 2, 3, . . . , 9}. Let A be the event that at least one of the tickets drawn is labeled with an even number, let B be the event that at least one of the tickets drawn is labeled with a prime number (recall that the number 1 is not regarded as a prime number). Suppose the numbers on the tickets drawn are 3 and 9.

Which of the following events occur?

1. $A$

2. $B$

3. $A$ and $B$ ($A \cap B$)

4. $A$ and $B^C$

5. $A^C$ and $B$

03:00
02:00

The Houston Astros beat the Philadelphia Phillies in the 2022 World Series. The winners in the World Series have to win a majority of 7 games, so the first team to win 4 games wins the series (best of 7). The Astros were heavily favored to win, so the outcome wasn’t really a suprise. Suppose we assumed that the probability that the Astros would have beaten the Phillies in any single game was estimated at 60%, independently of all the other games, what was the probability that the Astros would have won in a clean sweep?

(Clean sweep means that they won in the first 4 games - which didn’t happen, they won in 6 games.)

01:00

Suppose we assume, instead, that the probability that the Astros would have beaten the Phillies in any single game was 50%, independently of all the other games. In this case, was the probability that the series would have gone to 6 games higher than the probability that the series would have gone to 7 games, given that 5 games were played?

01:00

Let’s play a game where I first roll a fair six-sided die, and then toss a coin as many times as the number of spots I rolled. I win the game if I get all heads.

Given I roll a k what is the probability that I flip all heads?

03:00

Let’s play a game where I first roll a fair six-sided die, and then toss a coin as many times as the number of spots I rolled. I win the game if I get all heads.

What is the probability I win the game?

03:00

A rare condition affects 0.2% of the population. A test for this condition is 99% accurate: this means that the probability that a person with the condition tests positive is 99% and the probability that a person without the condition tests negative is 99%. What is the probability that a person who tests positive has the condition?

## Bayes’ theorem

Let $A$ and $B$ be events with positive probability. Then:

1. $B$ can be written as $B = (B\cap A) \cup (B \cap A^C)$

(ii)\begin{align} P(A|B) &= \displaystyle \frac{P(A \cap B)}{P(B)} \\ &= \frac{P(A \cap B)}{P(B\cap A) + P(B \cap A^C)} \\ &= \frac{P(B|A)P(A)}{P(B|A)P(A) + P(B|A^C)P(A^C)} \end{align}

# Break

05:00

# PS 8

25:00