Probabilities of Events

You should think of the probability of an event

A ⊆ Sas the weight of the subset A relative to the weight of the current universe S (it follows that the probability of S is 1)In this light the following formulas are kind of obvious:1 P(A B) = P(A) + P(B) – P(A∩B)∪2 P(Ac) = 1 – P(A)

Formula 2 is obvious because A and Ac add up to S.Formula 1 P(A B) = P(A) + P(B) – P(A∪ ∩B)is equally obvious from the figure

Since the weight of A∩B is counted twice inP(A) + P(B)

Formula 1 is called by our textbookThe Additive Rule of Probability

Other books call itThe Inclusion/Exclusion Principle

Whatever we call it, I prefer remembering it symmetrically as

P(A B) + P(A∪ ∩B) = P(A) + P(B)(I don’t have to remember where the minus sign goes!)

Any problem dealing with two events must give you enough information to determine, maybe using also the Rule of Complements P(Ac) = 1 – P(A)

three of the four numbers

P(A B), P(A∪ ∩B), P(A) and P(B)You compute the fourth one and fill the four spaces in the Venn diagram

Important AdviceWhen the problem deals with two events

do not read the question!First fill the four spaces in the Venn diagram,

Then read and answer the question(s) !

If a problem deals with three eventsdo not read the question!

First fill the eight spaces in the Venn diagram,

(The numbering of the spaces is arbitrary)

Then read and answer the question(s) !

Conditional ProbabilityRecall that the probability P(A) of an event A ⊆ Scan be thought as the

percentage of S embodied by AThere are situations when we will be interested in determining what

percentage of B is embodied by Afor some given event B. (instead of S)

Here are a couple of simple examples:

Toss a pair of fair dice. Let

A = the sum is evenB = the sum is 7 or less.The figure below shows that P(A) = = (the percentage of S embodied by A) =

But now we ask:

what percentage of B is embodied by A ?(In the language of gamblers, betting on A gives a fifty-fifty chance of winning, but should you change your bet if you are told that B happened?)The figure in the next slide shows B in celeste, with those entries of A which are part of B highlighted(larger and embossed.)

what percentage of B is embodied by A ?(COUNT !)

That’s right, , less than !! (change your bet!)

Here is another example. The table in the next slide shows the number, type and country of manufacture of the vehicles parked in a local WalMart parking lot yesterday at noon.Let

A = the vehicle is of foreign manufactureand

B = the vehicle is a passenger car

I assert P(A) = and

Percentage of B embodied by A =

The two examples suggest that:We need a symbol for

percentage of B embodied by Aand

What has this got to do with probabilities?We have encountered the symbol already, and the reason that the notion of probability is inherent here is that, focusing entirely on the event B we are asking the question

what are the chances of Aif B is our new universe?

Pictorially we are changing our universe fromto

Therefore in terms of relative weight,

the percentage of B embodied by AIs simply

the weight of A∩B relative tothe weight of B

i.e. (remember?)

As for the symbol(next slide …)

the percentage of B embodied by Ais written thus

P(A|B)and read

the probability of A given B

P(A|B) =

Now do all the homework assigned.