Math 30-2

Probability & Odds

Acceptable Standards(50-79%)

The student can express odds for or odds against as a probability determine the probability of an event determine odds for or odds against an event determine the odds for an event given the odds against the event and vice versa distinguish between mutually exclusive events and non-mutually exclusive events determine P(A or B) for events that are mutually exclusive determine P(A) when given P(A or B) and P(B) for mutually exclusive events interpret a model that represents any combination of mutually exclusive and non-

mutually exclusive events identify events that are complementary describe the elements that belong to the complement of a simple event determine the probability of the complement of an event, given the probability of the

event create a sample space using a graphic organizer distinguish between independent and dependent events determine P(A and B) for independent events determine P(A and B) for dependent events, given the order of the events

Standards of Excellence(80% +)

The student can also express probability as odds for or odds against provide an explanation for the validity of a probability statement provide an explanation for the validity of an odds statement determine P(A or B) for events that are non-mutually exclusive determine P(A) when given P(A or B), P(A and B), and P(B) for non-

mutually exclusive events represent events that are non-mutually exclusive using a graphic

organizer describe the elements that belong to the complement of a compound

event determine P(A) when given P(A and B) and P(B) for independent

events determine P(A and B) for dependent events when the order of the

events is not given

Vocabulary Probability (P) – is the likelihood that an event

will occur.

Outcomes – when you do a probability experiment the different possible results are called outcomes

Event – is a collection of outcomes

Sample space: the set of all possible outcomes. We denote S

Listing the Sample Space

Use a tree diagram to list the sample space for tossing a coin and rolling a die.

Coin Die Outcomes

H

123456

123456

T

H, 1H, 2H, 3H, 4H, 5H, 6

T, 1T, 2T, 3T, 4T, 5T, 6

The SampleSpace

Types of Probability There are 2 types of probability

Theoretical ProbabilityExperimental Probability

Let’s look at each one individually…

Theoretical Probability Theoretical Probability is based upon the

number of favorable outcomes divided by the total number of outcomes

Example: In the roll of a die, the probability of getting an

even number is 3/6 or ½.

Theoretical Probability Formula:

Example # 2

A box contains 5 green pens, 3 blue pens, 8 black pens and 4 red pens. A pen is picked at random

What is the probability that the pen is green?There are 5 + 3 + 8 + 4 or 20 pens in the box

P (green) = # green pens = 5 = 1 Total # of pens 20 4

Experimental Probability As the name suggests, Experimental

Probability is based upon repetitions of an actual experiment.

Example: If you toss a coin 10 times and record that the number of times the result was 8 heads, then the experimental probability was 8/10 or 4/5

Experimental Probability Formula:

P = Number of favorable outcomes Total number trials

Example In an experiment a coin is tossed 15

times. The recorded outcomes were: 6 heads and 9 tails. What was the experimental probability of the coin being heads?

P (heads) = # Heads = 6 Total # Tosses 15

The sum of all the probabilities of an event is equal to 1.

If P = 1, then the event is a certainty.If P = 0, then the event is impossible.

In probability, if Event A occurs, there is also the probability that Event A will not occur. Event A not occurring is the compliment of Event A occurring.The probability of Event A not occurring is written as P(A).(This is read as “Probability of not A”).

For Event A: P(A) + P(A) = 1

P(A) = 1 - P(A)

Complementary

One card is drawn from a deck of 52 cards. What is the probability of each of these events?a) drawing a red four b) not drawing a red four

a P red) ( )41

26 b P not red) ( )4 1

126

2526

Example

Odds

Odds Another way to describe the chance of an event occurring is

with odds. The odds in favor of an event is the ratio that compares the number of ways the event can occur to the number of ways the event cannot occur.

We can determine odds using the following ratios:

Odds in Favor = number of successes number of failures

Odds against = number of failuresnumber of successes

Also can write it as:odds in favor of A = number of outcomes for A : number of outcomes against A

Example Suppose we play a game with 2 number cubes.

If the sum of the numbers rolled is 6 or less – you win!

If the sum of the numbers rolled is not 6 or less – you lose

In this situation we can express odds as follows:

Odds in favor = numbers rolled is 6 or less numbers rolled is not 6 or less

Odds against = numbers rolled is not 6 or less numbers rolled is 6 or less

Example #2 A bag contains 5 yellow marbles, 3 white marbles,

and 1 black marble. What are the odds drawing a white marble from the bag?

Odds in favor = number of white marbles 3number of non-white marbles 6

Odds against = number of non-white marbles 6number of white marbles 3

Therefore, the odds for are 1:2 and the odds against are 2:1

PracticeSuppose you are watching a game show on TV. Five green doors are shown. Contestants on the show get to choose a door and potentially win a prize. Prizes can be found behind two of the five doors.

1) Determine the probability of winning a prize.2) Determine the probability of not winning a prize.3) Add the probability of winning and not winning a prize. What

do you notice?4) Use the following formula to write the odds as a fraction.

5) Write the odds of winning as a ratio. odds in favour of A = number of outcomes for A : number of outcomes against

6) Write the odds against winning as a ratio.

Discussion How does the probability of winning a prize compare to the

odds of winning a prize? What similarities and differences do you notice between

expressing the probability and the odds for an event? What do you notice about the odds for winning versus the

odds against winning? What relationship do you see?

Practice 2There is a 10% probability of winning a free play in the charity draw.1)Write 10% as a fraction.2)If 100 tickets are purchased, theoretically how many tickets would win a free play?3)If 100 tickets are purchased, theoretically how many tickets would not win a free play?4)Based on a 100 tickets being sold, what are the odds in favour of winning a free play? Write your answer as a ratio; then write it as a reduced ratio.5)Describe in words the information you can gain from knowing the probability of an event and how this information helps you write the odds for the event.

Probability vs Odds Probability is based on winning a prize out of all of the

possibilities, whereas odds are based on winning a prize compared to not winning a prize.

The difference between odds and probability is this: Probability is based on favourable outcomes in relation

to the total number of possible outcomes. Odds are based on the favourable outcomes “for” in

relation to unfavourable outcomes “against.”

Classifying Exclusivity

Two events are mutually exclusive if they cannot occur simultaneously. For instance, the events of drawing a diamond and drawing a club from a deck of cards are mutually exclusive because they cannot both occur at the same time.

For mutually exclusive events:

P(A B) = P(A) + P(B)

Mutually Exclusive:

A B

diamond club

Classifying Exclusivity

Events that are not mutually exclusive have some common outcomes. For instance, the events of drawing a diamond and drawing a king from a deck of cards are not mutually exclusive because the king of diamonds could be drawn, thereby having both events occur at the same time.

For events that are not mutually exclusive:P(A B) = P(A) + P(B) - P(A and B)

Non- Mutually Exclusive

Venn Diagram:Events that are Not Mutually

Exclusive

KingDiamonds

Both king and a diamond

These events are not mutually exclusive as it is possible for a card to be both king and a diamond

Classify each event as mutually exclusive or not mutually exclusive.

a) choosing an even number and choosing a prime number

b) picking a red marble and picking a green marble

c) living in Edmonton and living in Alberta

d) scoring a goal in hockey and winning the game

e) having blue eyes and black hair

Classifying Exclusivity

1. A box contains six green marbles, four white marbles, nine red marbles, and five black marbles. If you pick one marble at a time, find the probability of picking

a) a green or a black marble.

b) a white or a red marble.

Example

Determine the probability of choosing a diamond or a face card from a deck of cards.

Example 2

A national survey revealed that 12.0% of people exercise regularly, 4.6% diet regularly, and 3.5% both exercise and diet regularly. What is the probability that a randomly-selected person neither exercises nor diets regularly?

Example 3

Work on Practice Questions 1-8

P(A and B) = P(A) x P(B) (INDEPENDENT EVENTS)

Independent Versus Dependent Events

Two events are independent if the probability that each event will occur is not affected by the occurrence of the other event.

If the probabilities of two events are P(A) and P(B) respectively, then the probability that both events will occur, P(A and B), is:

If A and B are events from an experiment, the conditional Probability is the probability that Event B will occur given that Event A has already occurred. (dependent event)

Conditional Probability

P(B|A) is the notation for conditional probability. It should be read as “the probability of event B happening, given that event A has already occurred.”

A tree diagram is useful for modeling this types of problems

PracticeTwo events are dependent if the outcome of the second event isaffected by the occurrence of the first event.

Classify the following events as independent or dependent:

a) tossing a head and rolling a six

b) drawing a face card, and not returning it to the deck, and then drawing another face card

c) drawing a face card and returning it to the deck, and then drawing another face card

A cookie jar contains 10 chocolate and 8 vanilla cookies. If the first cookie drawn is replaced, find the probability of:

a) drawing a vanilla and then a chocolate cookie

b) drawing two chocolate cookies

Example

Find the probability of drawing a vanilla and then drawing a chocolate cookie, if the first cookie drawn is eaten.

Example

Determine the conditional probability for each of the following:

a) Given P(A and B) = 0.725 and P(A) = 0.78, find P(B|A).

b) Given P(blonde and tall) = 0.5 and P(B|A) = 0.68, find the P(blonde).

Practice

The local hockey time is having a raffle to raise money. The team is selling 2500 tickets, and there will be two draws. The first draw is for the grandPrize—a trip for two to an all-inclusive resort. The second draw is for the consolation prize-an HDTV. After each draw, the winning ticket is not return to the raffle. You buy 10 tickets for the raffle. Calculate the probability of winning the HDTV.What is the probability of winning at least one prize?

Finding Conditional Probability

Finding Conditional ProbabilityA diagnostic test for liver disease is accurate 93% of the time, and 0.9% of the population actually has liver disease.

a) Determine the probability the patient tests positiveb) Determine the probability the patient tests negative c) Determine the probability the patient has liver

disease and tests positived) Determine the probability the patient does not have liver disease and tests negative