PROBABILITY SUMMARY

Combining Events

If E and F are events In an experiment,

then:

E ∪ F is the event that either E occurs or

F occurs (or both).

E ∩ F is the event that both E and F occur.

E and F are said to

be disjoint or mutually exclusive if

(E F) is empty.

EC is the event that E does not occur.

Complement Rule:

P(Ec)= 1- P(E)

COMPLEMENTARY

CALCULATE PROBABALITY

ADDITION RULES

P (E U F) = n(E U F)

n(S)

OR

METHOD 2 METHOD 1

CONDITIONAL PROBABILITY

If E and F are two events, then

the conditional probability, P(E|F), is

the probability that E occurs, given that F

occurs.

MULTIPLICATION RULES

Example: 3 coins is thrown simultaneously. Find the probability

that heads comes up exactly once, given that the first comes up

heads

Let S be the original sample space for the experiment above;

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

Let E be the event that heads comes up at exactly once;

E = {HTT, THT, TTH}

and let F be the event that the first coin comes up heads;

F = {HHH, HHT, HTH, HTT}.

Then:

4/8

OR

n(S)=8

n(F)=4

Bayes' Theorem

Given two dependent events A and B, the

previous formulas for conditional

probability allow one to find P(A and B)

or P(B|A).

The rule is known as Bayes’ theorem.

It tells you “how to compute P(A|B) if you

know P(B|A) and a few often things.”

Bayes’s FORMULA

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

P(B)

EXAMPLE

Suppose that you are diagnosed with

microscopic hematuria .This symptom

occurs in 10 percent of all people and 100

percent of people with kidney cancer. You

would like to know the probability that you

have kidney cancer, which occurs in

0.000002 percent of all people.

EXAMPLE

SOLUTION:

P(A|B) = [P(B|A)][P(A)]

P(B)

=[1.0][0.000002]

[0.1]

= .00002

EXERCISES

EXERCISES

EXERCISES