Lecture 25 - Introduction
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Transcript 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?