Topics Conditional probability The AND (intersection) of two events Probability tables Probability trees

Experiment: I flip a coin twice E1 = two heads = {HH} P(E1) = ¼

suppose I tell you that the outcome was the same for both flips, that is, that event

E2 = {HH, TT} has occurred

Question: what is the probability that E1 occurred GIVEN THAT (you already know that) E2 has occurred? Answer: (can you think what it would be?) ··

This is called a conditional probability. Here's the mathematical notation:

We can investigate conditional probability using (you guessed it!): ··:

The KEY to conditional probability

P(E1 GIVEN E2) = P(E1 | E2) = = ··/··

But we can do it this way: P(E1 GIVEN E2) = P(E1 | E2) =

Rewriting the last fraction above (divide numerator and denominator by n(S)):

= =

So we have the formula:

P(A | B) =

Again, we have the probability of a compound event in terms of the probabilities of set combinations of its simple events.

So for our specific example, P(E1 | E2) = = 1/2, same answer as before!

These "clever" mathematical ways of formulating probabilities of compound events in terms of the probabilities of the constituent events constitute important mathematical tools for use in the study and calculation of probabilities.

Random experiment: flip coin twice Event E1: get head on first throw = {HH, HT} Event E2: get the same on both throws = {HH, TT}

Question: What is the probability of event E3 = E1 AND E2 that is, getting a head on first throw AND getting the same on the two throws?

Way 1: compute the event E3 from its definition. E3 = ··

Given events A and B, the AND of those two events is simply another event consisting of all outcomes that are in both A and B

Definition of A AND B

Here goes:

And we have the formula:

P(A B) = P(A)P(B | A) = P(B)P(A | B)

Probabilities for various situations are frequently displayed in a standard tabular form, as follows. Such tables often start as tables of counts, like the following classification of 200 adults according to gender and educational attainment:

This can be converted to an empirical probability table by dividing by 200:

P(college) = ·· P(college | female) = P(college AND female)/P(female) = ·· / ·· = ·· P(college | male) = P(college AND male)/P(male) = .11/.44 = .25 Notice that a conditional probability is (a joint probability) ÷ (a marginal probability)

A box contains 3 blue and 2 white balls. Two balls are drawn in succession, without replacement. Find the probability of drawing a white ball on the second draw.

Event notation:

P(W2) = 1/10 + 3/10 = 4/10 = 2/5

P(B2) = ·· + ·· = ·· = 3/5 (naturally!)