1.1 Theoretical Probability I

Introduction to Probability

Probability is the mathematics of chance. There are three basic approaches:

o Experimental Probability: is based on the results of previous observations. Experimental probabilities are relative frequencies and give an estimate of the likelihood that a particular event will occur.

o Theoretical Probability: is based on the mathematical laws of probability. It applies only to situations that can be modelled by mathematically fair objects or experiments.

o Subjective Probability: is an estimate of the likelihood of an event based on intuition and experience making an educated guess using statistical data.

Fair Game: A game is FAIR if: All players have an equal chance of winning or Each player can expect to win or lose the same number of times

in the long run.

A trial is one repetition of an experiment

An event is a possible outcome of an experiment.

A simple event is an event that consists of exactly one outcome.

1. Experimental Probability:o Is based on the data collected from actual experiments

involving the event in question.

o An experiment is a sequence of trials in which a physical occurrence is observed

o An outcome is the result of an experiment

o The sample space is the set of all possible outcomes

o An event is a subset of the sample space - one particular outcome

Example 1:

1.1 Theoretical Probability I

Suppose you flipped a coin 30 times and, tails showed 19 times. The outcomes are H or T, and the event E = tails.

Example 2:

If you rolled two dice 20 times and a total of 7 showed up three times. Then,

2. Theoretical Probability:

It is based on the mathematical laws of probability. It applies only to situations that can be modeled by mathematically fair objects or experiments.

Assumes that all outcomes are equally likely

The probability of an event in an experiment is the ratio of the number of outcomes that make up that event over the total number of possible outcomes

Let the probability that an event A occurs be P(A) then,

where n(A) is the number of times event A happens and n(S) is the number of possible outcomes in the sample space.

For n outcomes, the sum of probabilities of the outcomes is 1.

P1 + P2 + P3 +...+Pn = 1i. Simple Probability:

Example 1:

Rolling one die: Sample space = {1, 2, 3, 4, 5, 6}

a) If event A = rolling a 4 then

b) If event B = rolling an even number then

Example 2:Suppose a bag contains 5 red marbles, 3 blue marbles and 2 white marbles.

1.1 Theoretical Probability I

If event A = drawing out a blue marble then,

Total number of Blue marbles: 3

Total number of marbles: 10

Example 3:

Calculate the probability of rolling a sum of 7 when rolling two six-sided dice.

Event A = rolling a sum of 7

P (A )= 636

P(A) = 16

ii. Tree Diagrams

Sam and Julia want to have three children. What is the probability that they will have 2 boys and 1 girl? Draw a tree diagram to represent each outcome.

1st Child 2nd Child 3rd Child Outcomes:

Girl

Boy

P(2 boys and 1 girl) = 3/8

Verify that the sum of the probabilities of the outcomes is equal to1:iii. Weighted Trees

Girl

Boy

Girl

Girl

Girl

Girl

GirlBoy

Boy

Boy

Boy

Boy

1.1 Theoretical Probability I

o Each outcome in the sample space is not equally likelyo Such models can be easily represented by tree diagrams

Example 1:

a) 1600 ants start climbing a tree as shown in the diagram below. At each branch, ¼ of the ants go to the left and ¾ go to the right.

a. Along each branch, write the probability that an ant at the bottom of the branch moves up that branch.

b. What is the probability that a particular ant reaches v?

Event A = Ant reaches V

P(A) =

Example 2:

0.25 0.75

0.250.25

0.250.25

0.750.75

0.75 0.75 0.250.75

1.1 Theoretical Probability I

a) What is the probability of a pass on a 4 question multiple choice test where each question has 4 possible answers?

Method 1: Chart

Method 2: Tree Diagram

b) Monica estimates that the probability of getting the next question right if the previous question was right on a Data Management test is ¾. But

Q#1 Q#2 Q#3 Q#4R R R RR R R WR R W RR W R RW R R RR R W WR W R WW W R RW R R RW W W RW W R WW R W WR W W WW W W WW R W R[

1.1 Theoretical Probability I

the probability of getting it right if the previous one was wrong is only 1/3. The probability of getting the first question right is 4/5.

a. Draw a tree diagram showing the probabilities of getting 3 questions right or wrong.b. What is the probability that she gets the third question right?c. What is the probability of a pass?

iv. Probability with/without replacement

There are 4 Blue blocks, 5 Red blocks and 3 Green blocks in a bag.

a. Find the probability of selecting 2 Blue blocks with replacement

b. What is the probability of selecting 2 Blue blocks without replacement?

v. Complementary Events

The complement of a set A is written as A’ and consists of all the outcomes in the sample space that are NOT in A.

Example 1:

Rolling one die: Sample space = {1, 2, 3, 4, 5, 6}

If event A = rolling a 4 then and A’ = not rolling a 4 then,

Generally: P(A’) = 1 – P(A)

The minimum value for any probability is 0 (impossible)

The maximum value for any probability is 1 (certain)

Probability can be expressed as a ratio, a decimal or a percent

1.1 Theoretical Probability I

1.1 Theoretical Probability I