# STAT 1450 Lecture Notes - Lecture 13: Probability Theory, Venn Diagram

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Published on 17 Sep 2020

Department

Statistics

Course

STAT 1450

Professor

Chapter 13, page 1

STAT 1450 COURSE NOTES – CHAPTER 13

GENERAL RULES OF PROBABILITY

Connecting Chapter 13 to our Current Knowledge of Statistics

Probability theory leads us from data collection to inference.

The introduction to probability from Chapter 12 will now be fortified by additional rules to allow

us to consider multiple types of events. The rules of probability will allow us to develop models

so that we can generalize from our (properly collected) sample to our population of interest.

13.1 Independence and the Multiplication Rule

Two events A and B are independent if knowing that one occurs does not change the probability

that the other occurs. Thus, if A and B are independent,

Example: Someone with type O-negative blood is considered to be a “universal donor.

According to the American Association of Blood Banks, 39% of people are type O-negative.

Two unrelated people are selected at random. Calculate the probability that both have type

O-negative blood.

Poll: Suppose two unrelated people are selected at random.

Calculate the probability that neither have type O- negative blood

(round to 4 decimal places).

PCAandBPCA PCB

PCABPco

39 xPC

0.39 0.1521

Chapter 13, page 2

13.2 The General Addition Rule

Two-Way tables are helpful ways to picture two events.

Venn diagrams are an alternative means of displaying multiple events.

Both can be used to answer many questions involving probabilities.

Example: In a sample of 1000 people, 88.7% of them were right-hand dominant, 47.5% of them

were female, and 42.5% of them were female and right-hand dominant. Draw a Venn diagram

for this situation.

Calculate the probability that a randomly selected person is right-hand dominant or female.

We just used the general addition rule:

For any two events A and B,

Question: Where did we see this concept previously?

What if the two events A and B do not overlap?

Events A and B are ________________ if they have no outcomes in common.

Question: What is P(A or B) when A and B are disjoint?

Example: Calculate the probability that a randomly selected person is neither right-hand

dominant nor female.

We just used the complement rule: For any event A, P(A does not occur) = P(not A) = 1 – P(A).

0.462

0.063

0.425

0.050

PCR PcM RHPCFdR RF

0.46210.425

0.8871 47.5 42.5 0.937

PCAorB PCA 11713 pcadB OR

disjoint

PCAandBonooutcomesin common

PCAorB _PCA fPCB PCA Bgeneral

addition

rule

pA1

PCB OC

disjoint

PChotF RlPCForR l0.937 0.063

## Document Summary

Connecting chapter 13 to our current knowledge of statistics. Probability theory leads us from data collection to inference. The introduction to probability from chapter 12 will now be fortified by additional rules to allow us to consider multiple types of events. The rules of probability will allow us to develop models so that we can generalize from our (properly collected) sample to our population of interest. Two events a and b are independent if knowing that one occurs does not change the probability that the other occurs. Example: someone with type o-negative blood is considered to be a universal donor. (cid:180) According to the american association of blood banks, 39% of people are type o-negative. Poll: suppose two unrelated people are selected at random. Calculate the probability that neither have type o- negative blood (round to 4 decimal places). Two-way tables are helpful ways to picture two events. Venn diagrams are an alternative means of displaying multiple events.