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Probabilities of Events. You should think of the probability of an event A ⊆ S a s 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:

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probabilities of events
Probabilities of Events

You should think of the probability of an event

A ⊆ S

as 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:

  • P(A∪B) = P(A) + P(B) – P(A∩B)
  • P(Ac) = 1 – P(A)
slide2

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 in

P(A) + P(B)

slide3

Formula 1 is called by our textbook

The Additive Rule of Probability

Other books call it

The 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)

slide4

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 advice
Important Advice

When 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) !

slide6

If a problem deals with threeevents

do not read the question!

First fill the eightspaces in the Venn diagram,

(The numbering of the spaces is arbitrary)

Then read and answer the question(s) !

conditional probability
Conditional Probability

Recall that the probability P(A) of an event A ⊆ S

can be thought as the

percentage of S embodied by A

There are situations when we will be interested in determining what

percentage of B is embodied by A

for some given event B. (instead of S)

Here are a couple of simple examples:

Toss a pair of fair dice. Let

slide8

A = the sum is even

B = the sum is 7 or less.

The figure below shows that P(A) =

= (the percentage of S embodied by A) =

slide9

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.)

slide11

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 manufacture

and

B = the vehicle is a passenger car

slide12

I assert P(A) =

and

Percentage of B embodied by A =

slide13

The two examples suggest that:

We need a symbol for

percentage of B embodied by A

and

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 A

ifBis our new universe?

slide15

Therefore in terms of relative weight,

the percentage of B embodied by A

Is simply

the weight of A∩B relative to

the weight of B

i.e. (remember?)

As for the symbol

(next slide …)

slide16

the percentage of B embodied by A

is written thus

P(A|B)

and read

the probability of A given B

P(A|B) =

Now do all the homework assigned.