3.1 & 3.2: Fundamentals of Probability

Objective: To understand and apply the basic probability rules and theorems

CHS Statistics

Inferential Statistics• Recall: Inferential statistics involve making predictions

about a population, given a sample. In probability we use population information to infer the probability.

Key Definitions

• Theoretical Probability – what the probability should be

• Ex) The probability of flipping a coin and getting a tails should always be ½ or 0.5.

• Experimental Probability – the actual probability found after an experiment

• Ex) You got 6 tails after flipping a coin 8 times, so your experimental probability is 6/8 or 0.75.

Key Definitions (continued)• Event – a trial of an experiment

• Often notes as P(A) – “the probability of Event A”• Ex) flipping a coin

• Outcome – result of a single trial• Ex) not getting tails

• Simple event – an outcome or event that cannot be further broken down into simpler components.• Ex) The outcome of ‘5’ after rolling one die.• Ex) The outcome of ‘8’ as the sum of rolling two dice

is not a simple event, because there are various ways to get 8 (4-4, 5-3, 2-6).

Key Definitions (continued)• Sample Space – the set of all possible simple event

outcomes

• Ex) The sample space for flipping a coin is: {H, T}

• Ex) The sample space for rolling a die is:

• Ex) The sample space for flipping two coins is:

Notation for Probabilities• P denotes a ______________.

• A, B, C, etc. denote ________________.

• P(A) denotes _________________________________.

Calculating Probabilities

• Example of an event that will never occur: 

• Example of an event that will always occur:

Law of Large Numbers

• The more an experiment is repeated, the closer the experimental probability will get to the theoretical probability.

• Virtual Representation of Law of Large Numbers

Basic Probability Examples:

1) On an SAT and ACT test, a typical multiple choice question has 5 possible answers. If you make a random guess on one such question, what is the probability that your response is wrong?

2) Consider the SAT and ACT test again, but now consider that a 1/5 point penalty is assessed for incorrect answers and 1 point is assessed for correct answers. You are not penalized for skipping questions. If there are 5 answers to choose from and 20 questions, should you guess?

Basic Probability Examples (continued):

3) A couple has 3 children. Find the probability that: 

 • they will have exactly 2 boys

 • they will have at least 1 boy

 • they will have no boys

 

Complementary Events• The complement of event A, denoted by , consists of all

outcomes in which event A does not occur.

P() = 1 – P(A)

• Example: The General Motors Corporation wants to conduct tests of a new model of the Corvette. A pool of 50 drivers has been recruited, 20 of whom are men. When the first person is selected from this pool, what is the probability of not getting a male driver?

Fundamental Counting Principle• If one event occurs m ways and a second event occurs n

ways, the number of events that both events can occur is (m)(n)

• Example: The access code for a car’s security system consists of four digits. How many codes are possible if:• Each code can be used only once and not repeated?

• Each digit can be repeated?•  

• Each digit can be repeated but the first digit cannot be 0 or 1?

Odds• Odds Against – The actual odds against event A are the

ratio P() : P(A) in reduced form.

• Odds in Favor – The actual odds in favor of event A are the reciprocal of the odds against that event, P(A) : P() in reduced form.

• Payoff odds – The payoff odds against A represent the ratio of the net profit (if you win) to the amount bet.

Payoff odds against event A = (net profit) : (amount bet)

Odds Examples

• What are the odds against rolling a 5 when a single die is rolled?

   

Odds Examples (continued)If you bet $5 on the number 13 in roulette, your probability of winning is 1/38 and the payoff odds are given by the casino 35:1.• Find the actual odds against the outcome of 13.  • How much net profit would you make if you win by betting 13?  • If the casino were operating just for fun of it, and the payoffs odds were

changed to match the actual odds against 13, how much would you win if the outcome were 13?

   

Assignment:• pp. 123 – 127

# 1, 2, 6 – 10 EVEN, 16 – 20 EVEN, 31, 35