# Computing Probabilities

STAT 20: Introduction to Probability and Statistics

## Agenda

• Announcements
• PS 6
• Concept Review
• Concept Questions
• Break
• PS 7 (computing probabilities)
• Break

## Announcements

• Problem Sets 6 and 7 (paper, max. 3) due Tuesday at 9am
• No lab this week.
• RQ: Probability Distributions due Mon/Tues 11:59pm

# Concept questions & review

## Rules

• Conditional Probabilty

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

• Multiplication rule

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

• Complement rule

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

## Concept Question 1

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: ________

## Concept Question 2

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: ________

## Concept Question 3

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.)

## Concept Question 5

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?

# Break

05:00

# PS 7

25:00