Download Lecture 25 - Introduction

Introduction
Lecture 25
Section 6.1
Wed, Mar 22, 2006
What is Probability?
A coin has a 50% chance of landing heads.
 What does that mean?


The coin will land heads 50% of the time?
• This is demonstrably false.

The coin will land heads approximately
50% of the time?
• Then the probability is approximately 50%, not
exactly 50%.
The Meaning of Probability
It means that the fraction of the time that the
coin lands heads will get arbitrarily close to
50% as the number of coin tosses increases
without bound.
 This involves the notion of a limit as n
approaches infinity.
# heads 1
lim

#tosses # tosses
2

The Sample Space
An experiment is a procedure that leads to
an outcome.
 If at least one step in the procedure is left
to chance, then the outcome is
unpredictable.
 We observe a characteristic of the
outcome.
 The sample space is the set of all possible
observations.

The Sample Space

Example
Procedure: Toss a coin.
 Observed characteristic: Which side landed
up.
 Sample space = {H, T}

The Sample Space

Example
Procedure: Roll a die.
 Observed characteristic: Which number
landed up.
 Sample space = {1, 2, 3, 4, 5, 6}

Calculation of Probability
We will consider only finite sample spaces.
 If the n members of the sample space are
equally likely, then the probability of each
member is 1/n.
 Examples

Toss a coin, P(H) = 1/2.
 Roll a die, P(3) = 1/6.

The Probability of an Event
An event is a collection of possible
observations, i.e., a subset of the sample
space.
 The probability of an event is the sum of
the probabilities of its individual members.
 If the members of the sample space are
equally likely, then P(E) = |E|/|S|.

Example: Probability of an
Event
In a full binary search tree of 25 values,
what is the probability that a search will
require 5 comparisons?
 Assume that all 25 values are equally
likely.
 10 of them occupy the bottom row.
 Therefore, p = 10/25 = 40%.

Example
A deck of cards is shuffled and the top card
is drawn.
 What is the probability that it is

The ace of spades?
 An ace?
 A spade?
 A black card?

Example
A deck of cards is shuffled, the top card is
discarded, and the next card is drawn.
 What is the probability that it is

The ace of spades?
 An ace?
 A spade?
 A black card?

Example
A deck of cards is shuffled, the top card is
drawn, and it is noted that it is red. Then
the next card is drawn.
 What is the probability that it is

The ace of spades?
 An ace?
 A spade?
 A black card?

Example
A deck of cards is shuffled, the top card is
drawn, and it is noted that it is black. Then
the next card is drawn.
 What is the probability that it is

The ace of spades?
 An ace?
 A spade?
 A black card?

Example
Two red cards and two black cards are laid
face down.
 Two of them are chosen at random and
turned over.
 What is the probability that they are the
same color?

The Monty Hall Problem
See p. 301.
 There are three doors on the set for a
game show. Call them A, B, and C.
 You get to open one door and you win the
prize behind the door.
 One of the doors has a Ferrari behind it.
 You pick door A.

The Monty Hall Problem
However, before you open it, Monty Hall
opens door B and shows you that there is a
goat behind it.
 He asks you whether you want to change
your choice to door C.
 Should you change your choice or should
you stay with door A?

The Monty Hall Problem

There are three plausible strategies.

Stay with door A.
• Door C still has a 1/3 chance, so door A must
have a 2/3 chance.

Switch to door C.
• Door A still has a 1/3 chance, so door C must
have a 2/3 chance.

It doesn’t matter.
• Both doors now have a 1/2 chance.
The Monty Hall Problem

Use a simulation to determine the correct
answer.

MontyHall.exe.
A Contest Problem

If we choose an integer at random from 1
to 1000, what is the probability that it can
be expressed as the difference of two
squares?