# STAT230 Study Guide - Final Guide: Random Variable, Probability Distribution, Continuity Correction

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10 Dec 2018

School

Department

Course

Professor

University of Waterloo

STAT 230

Probability

Fall 2018

Final Exam

Prof: Adam Kolkiewicz.

Exam Guide

Topics included:

1. Intro to probability

2. Probability Models

3. Counting techniques

4. Probability Rules and Conditional Probability

5. Discrete Random Variables

6. Expected Value and Variance

7. Continuous Random Variables

8. Multivariate Distributions

Chapter 1 – Intro to Probability

Definition

1. Classical definition: probability of some event is

2. Relative frequency:

- (limiting) proportion of times the event occurs in a very long series of

repetitions in the experiment

3. Subject probability definition

- A measure of how sure the person states something will happen.

Chapter 2 – Probability Models

Sec 1. Sample space

1. Sample space: the set of all possible(distinct) outcome of the random experiment

2. Set of all subsets of the sample space: “events” going to assign probabilities

3. Mechanism for probabilities: [0, 1].

Sec 2. Discrete probability distribution

1. For a discrete sample space containing ai, 0≤P(ai) ≤ 1. And

2. Uniform Probability Model: for the sample space S with equally likely simple

events, we have the probability of every individual outcome is the same.

Sec 3. “Odds”

Definition: The odds in favour of an event A is the probability with which the

event occurs divided by the probability it does not occur: assuming P(A) > 0.

The odds against the event is the reciprocal of the odds in favour.

Exercise questions

Question 1. Roll a six sided die.

a. What is the probability that the number is 6?

b. What is the probability that the number is less than 6?

c. What is the probability that the number is even?

Solution:

A = the rolled number is 6

B = the rolled number is less than 6

C = the rolled number is even

S = {1, 2, 3, 4, 5, 6}

a. Probability =

=

b. Probability =

=

c. Probability =

=

Question 2. Draw a card hand at random from a standard 52 card deck of

playing cards (13 kinds: A, 2, 3, 4,... 10, J, Q, K, in 4 suits).

i) What is the probability that the hand contains is diamond?

ii) What is the probability that the hand is smaller than 10?

Solution:

The events of interest:

A = the hand contains a diamond

B = the hand contains < 10

i) Probability =

=

= 1/4

ii) Probability =

=

= 9/13

Question 3. Draw a card hand at random from a standard 52 card deck of

playing cards (13 kinds: A, 2, 3, 4,... 10, J, Q, K, in 4 suits).

What is the odd in favour of choosing the diamond? And odd against?

Solution:

- If the probability that the selected card is diamond is ¼, then the odds in

favour of diamonds are ¼ / ¾ = 1/3. That is 1:3.

- So the odds against diamonds are 3:1.