Lecture 10

Chance

Summer 2022

Announcements

• HW 4 due tonight, 7/5
• Project 1 Checkpoint due tomorrow, 7/6
• Full project due Friday, 7/8
• Midterm Preference Form on Ed due 7/8
• Tutoring Sections Expanding! Today @ Noon
• Ellen’s OH today cancelled
• Makeup: 11:30am-12:30pm tomorrow, 7/6

Weekly Goals

• Last Week
• Simulating Probabilities
• Today
• Calculating Probabilities
• Finding Chances
• This Week
• Models that involve chance
• Assessing the consistency of the data and the model

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The Monty Hall Problem

Monty Hall Problem

https://probabilityandstats.files.wordpress.com/2017/05/monty-hall-pic-1.jpg

The Final Choice

(Demo)

https://en.wikipedia.org/wiki/Monty_Hall_problem

Stay or Switch?

https://probabilityandstats.files.wordpress.com/2017/05/monty-hall-pic-1.jpg

(Demo)

Finding Chances

Basics

• Lowest value: 0
• Chance of event that is impossible
• Highest value: 1 (or 100%)
• Chance of event that is certain

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• Complement: If an event has chance 70%, then the chance that it doesn’t happen is
• 100% - 70% = 30%
• 1 - 0.7 = 0.3

Equally Likely Outcomes

Assuming all outcomes are equally likely, the chance of an event A is:

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number of outcomes that make A happen

P(A) = ---------------------------------------------------------------

total number of outcomes

A Question

• I have three cards: ace of hearts, king of diamonds, and queen of spades.
• I shuffle them and draw two cards at random without replacement.

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• What is the chance that I get the Queen followed by the King?

(Demo)

Multiplication Rule

Chance that two events A and B both happen

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= P(A happens) x P(B happens given that A has happened)

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• The answer is less than or equal to each of the two chances being multiplied
• The more conditions you have to satisfy, the less likely you are to satisfy them all

Another Question

• I have three cards: ace of hearts, king of diamonds, and queen of spades.
• I shuffle them and draw two cards at random without replacement.

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• What is the chance that one of the cards I draw is a King and the other is Queen?

(Demo)

Addition Rule

If event A can happen in exactly one of two ways, then

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P(A) = P(first way) + P(second way)

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• The answer is greater than or equal to the chance of each individual way

Complement: At Least One Head

• In 3 tosses:
• Any outcome except TTT
• P(TTT) = (1/2) x (1/2) x (1/2) = 1/8
• P(at least one head) = 1 - P(TTT) = 1 - (1/8) = 87.5%

• In 10 tosses:
• 1 - (1/2)**10 ≅ 99.9%

(Demo)

Problem-Solving Method

Most problems involve multiple trials. Here’s a method that works widely.

• Ask yourself what the first trial has to be. If there’s a clear answer (e.g. “not a six”) whose probability you know, almost certainly you can continue the process with the multiplication rule.
• If there’s no clear answer (e.g. “could be R, could be B, but then the next one would have to be B, or R …”), list all the distinct ways your event could occur and add up their chances.
• If the list above is long and complicated, look at the complement. If the complement is simpler (e.g. the complement of “at least one” is “none”), you can find its chance and subtract that from 1.