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7.4
Basic Concepts of Probability
• This presentation is copyright by Paul Hendrick
• © 2003-2005, Paul Hendrick
• All rights reserved
7.4
Basic Concepts of Probability
• Union rule for probability
– Union rule (general case – always true)
P( E  F )  P( E )  P( F )  P( E  F )
– IF P(EF) = 0, it can be omitted!
– Union rule for mutually exclusive events, only
P( E  F )  P( E )  P( F )
7.4
Basic Concepts of Probability
Complement rule
– “back-door” approach
• (because it doesn’t calculate directly)
P( E )  P( S )  P( E )
P( E )  1  P( E )
P( E )  P( S )  P( E )
P( E )  1  P( E )
7.4
Odds
– Probability uses two numbers -- # ways for a successful outcome
& # ways for any outcome
•
•
•
•
•
n(E) and n(S)
P(E) = n(E) / n(S)
Let S={1,2,3,4,5,6} be the sample space for rolling a single fair die.
Let E={1,6} be the event of rolling an “extreme” number.
n(E) = 2; n(S) = 6;
so P(E) = 2/6 = 1/3.
– There is a third number -- # ways for an unsuccessful outcome
• n(E’)
• n(E’) = n(S) - n(E)
• For the above, n(E’) = 6-2 = 4,
the number of ways of NOT rolling an “extreme” number.
• Note: P(E’) = 4/6 = 2/3 or P(E’) = 1 – P(E) = 1-1/3 = 2/3.
7.4
Odds (cont)
– odds uses a different two of the three numbers
– odds in favor of an event E
• = n(E) / n(E’) (assumes all outcomes are equally-likely)
• Odds in favor of an “extreme” roll are 2 / 4 or 1/2
• = P(E) / P(E’) (uniform sample space NOT necessary for
this formula)
• Odds for “extreme” roll also by 1/3 / 2/3 = 1/2
– odds against an event E
• = n(E’) / n(E) (numerator and denominator switched!)
• Odds against an “extreme” roll are 4 / 2 or 2/1
– Again, note the 2 & 4 are reversed from the first example above.
7.4
Odds (cont)
– Instead of as fractions, odds are commonly shown as ratios with a
colon used to show comparison
– n(E) : n(E’) instead of n(E) / n(E’)
– Odds in favor of an “extreme” roll would then be
2:4 or 1:2 instead of 2/4 or ½
– (read as “two to four”, or “1 to 2”, resp.)
– Odds against an “extreme” roll would then be
4:2 or 2:1 instead of 4/2 or 2/1 or even just 2
– Note in the above, that odds are generally reduced, just as
fractions are.
7.4
Odds (cont)
– A lot of people confuse odds with probability -- they
are similar ideas (and sometimes close numbers), but
are not the same.
– Recapping the previous example of the event E = the
“extreme” roll of a die,
– P(E) = 2/6 = 1/3 ; odds for E are 2:4 or 1:2
– P(E’) = 4/6 = 2/3 ; odds against E are 4:2 or 2:1
– Different example, consider F = “rolling a sum of 12”
on two fair dice:
– P(F) = 1/36 ; odds for F are 1:35
– P(F’) = 35/36 ; odds against F are 35:1
7.4
Odds (cont)
– You should understand the similarities and also the
differences between “odds” and “probability”.
– You should be able to calculate both:
• Odds in favor of an event E (or simply “for” the event)
• Odds against an event E
– You should be able to convert from probabilities to
odds, or vice versa, on a given problem.
• The book gives some formulas for this on page 349, if you
want to do it “by formula”
– Odds are mainly used by gamblers for handling
money; we’re not too concerned with this in class
– We will predominantly use probabilities in class – in
fact combinations such as union and intersection are
much easier to do with probabilities than with odds!
7.4
Further probability notions
• Types of probability
– theoretical (by counting in a uniform sample space -- the formula
P(E) = n(E) / n(S) )
– empirical (by having observed typical outcomes -- an
experiment)
• In a city study at an intersection, out of the 500 northbound cars, 35
of them turned left. What’s the probability of such a car turning
left?
• The empirical probability = 35 / 500 = 7 / 100 or 7%
• This kind of probability is sometimes referred to as “relative
frequency”
– intuitive ( a “gut feeling” -- from experience?)
• You think you have a “fifty-fifty” chance of “acing” exam 1.
• A businessman who has a successful chain of 20 pizza restaurants
estimates a new restaurant on Texas at University will have an 85%
chance of being successful.
7.4
Further probability notions
• Probability distribution for an experiment
– Simply a list of all possible outcomes and their
associated probabilities
– Easiest given as a table
– Probability distribution for 1 fair die:
E
Pr
1
1/6
2
1/6
3
1/6
4
1/6
5
1/6
6
1/6
– Probability distribution for sum of 2 fair dice:
E
2
3
4
5
6
7
8
9
10
11
12
Pr
1/36
1/18
1/12
1/9
5/36
1/6
5/36
1/9
1/12
1/18
1/36
7.4
Further probability notions
• Properties of probability
– Let S be a sample space consisting of n distinct (i.e., mutually
exclusive) outcomes, s1, s2, … , sn. An acceptable probability
assignment consists of assigning to each outcome si a number pi
(the probability of si ) according to these rules:
– 1. The probability of each outcome is a number
between 0 and 1. (PINGTO! and PINN!)
• 0 <= p1 <= 1, 0 <= p2 <= 1, … , 0 <= pn <= 1,
– 2. The sum of the probabilities of all possible outcomes is 1.
• p1+ p2+ p3+ … + pn=1
• (or Spi  1, for short)
 3. Don’t forget:
P( E  F )  P( E )  P( F )  P( E  F )