Probability

What is probability?• Probability discusses the likelihood or chance of

something happening. For instance, • -- the probability of it raining tomorrow• -- the probability of rolling a 4 on a six-sided die• -- the probability of being struck by lightning• -- the probability of pulling an Ace out of a deck

of cards• We will try to answer some of these questions in these

chapters. Let’s start with a relatively easy example.

• Consider rolling a six-sided die. What is the probability of getting a 4 on the die? Notice the notation we will use.

61)4( P

We can think of this as the ratio of successes to total

possibilities. There is 1 success (the 4) on a die and

6 possible sides.

• First, this does assume all sides are equally likely to come up when I roll the die. This is usually the case when we talk about dice.

• This formula for probability will come in handy. Keep it in mind.

• We should cover some terminology that we will use throughout our work here. Examples are given after each term.

Terminology• experiment: the act you do (roll a die)• outcome: any one possible result (1, 2, 3, 4, 5, or 6)• sample space: the set of all possible outcomes • ( {1, 2, 3, 4, 5, 6} )• event: a particular set of outcomes ( {2, 4, 6} or evens)• trial: one instance of the experiment (If you roll a die

50 times, you’ve done 50 trials.)• success: specific event you’re interested in (If we want

the probability that the die will be even, we have three successes; they are 2, 4, and 6.)

• Theoretical probability: the probability of an event based on the context of the problem

• Experimental probability: the probability of an event based on doing the experiment many times

• Consider rolling a die. The theoretical probability of • getting an even number is • gotten by thinking about the die and figuring

that there are 3 successes and 6 total possibilities. • If we were to roll a die 100 times and actually roll an

even number 47 of the times, our experimental probability

• would be .

63)( evenP

10047)( evenP

Counting Sample points

As a further illustration, we may be interested in the event B that the number of defectives is greater than 1 in previous Example. This will occur if the outcome is an element of the subsetB = {DDN,DND,NDD,DDD} of the sample space S.To each event we assign a collection of sample points, which constitute a subsetof the sample space. That subset represents all of the elements for which the eventis true.

Venn Diagram

Consider the experiment

• Roll die, toss coin. • What’s P(4 and H)?

• Sample space: 1T 1H• 2T 2H• 3T 3H• 4T 4H• 5T 5H• 6T 6H

You need to write the outcomes down in an

orderly fashion.

Now calculate

• So P(4 and H) = 1/12. Let’s look at it a different way.

• Notice P(4 on die) = 1/6 • and P(H on coin) = ½.

• And

• This is an example of a commonly used rule. We’ll state it in general but we need a definition first.

)4()()4(

121

21

61

andHPHPP

There are 12 possibilities and 1 success…

Independent events• Two events are independent if the occurrence of one

does not affect the occurrence of the other. The events “4” and “H” are independent because the occurrence of the 4 on the die does not affect the probability that I will get an H also.

• Rule: If events E and F are independent, then the probability that they both occur is equal to the probability that E occurs times the probability that F occurs or

• .)()()( FPEPEandFP

Mutually exclusive events• Two events are mutually exclusive iff they cannot

happen at the same time.

• expl: experiment: roll die • A: roll even number• B: roll a 3

• Events A and B are mutually exclusive because you cannot roll an even number and a 3 at the same time.

Exhaustive Events• Events are said to be collectively exhaustive events

when the union of mutually exclusive events is the entire sample space S.

Worksheet• “Probability Worksheet ” covers the four types of problems

we will encounter. They are finding the probability that• 1.) either of two mutually exclusive events occur,• 2.) either of two non-mutually exclusive events

occur,• 3.) two independent events both occur, and• 4.) two non-independent events both occur.• Its last page contains some practice problems.• “Solutions to Probability worksheet ” is available. It

explains the practice problems at the end of the worksheet.

•There are four rules that are discussed on this worksheet. They are summarized here.

•1.) If A and B are mutually exclusive, • then P(A or B) = P(A) + P(B). •2.) If A and B are not mutually exclusive, • then P(A or B) = P(A) + P(B) – P(A and B).•3.) If A and B are independent,• then P(A and B) = P(A) P(B). •4.) If A and B are not independent, • then P(A and B) = P(A) P(B given A).

Some questions• 1.) What is the lowest a probability can be?

Can it be 0 or negative?• 2.) What is the highest a probability can be?

Can it be 2?• 3.) What is the sum of the probabilities of all

possible mutually exclusive events?

Use the experiment of rolling a die 100 times. What is the lowest P(4) can be?

What is the highest? What is P(1) + P(2) + P(3) + P(4) + P(5) + P(6)?

Some answers• Since experimental probability is ,

and

• since the number of successes cannot be negative but it could be zero, the lowest a probability can be is 0.

• Since the number of successes cannot exceed the number of trials, the highest a probability can be is 1.

• We see that P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1. Since these are the only six things that can occur and one must occur, then the sum of the probabilities of the possible mutually exclusive events must be 1.

Terminology• An event that has probability 1 must always happen. It

is called a sure or certain event.• experiment: Toss coin• event: heads or tails • When you toss a coin, you must get either a heads or a

tail.

• An event that has probability 0 will never happen. It is called an impossible event.

• experiment: Roll die• event: roll 7• When you roll a six-sided die, you cannot get a 7.

Complement of an event• If we let A represent an event, then is used to represent

its complement. • The complement of an event is made up of the outcomes

from the sample space that are not in the original event. • Consider pulling a single card out of a deck of poker cards.

Let A represent the event “red Queen”. Here, A can be interpreted as “the Queen of Hearts or the Queen of Diamonds”.

• The complement of A would be all of the other 50 cards. • Since either an event occurs or it does not, A and must

be mutually exclusive; they cannot happen at the same time. Since either A or must happen, we know

• .

A

A

A

( ) ( ) 1P A P A

A more complicated example• Consider the experiment of rolling two distinguishable six-

sided dice. We are interested in finding the probability that the sum of the two dice rolled is 6. Let’s look at the sample space (36 events) of this experiment.

• In finding the probability of rolling two dice whose sum is 6, we need to think about

• Since there are 5 successes out of 36 equally likely • possibilities, the probability of rolling a sum of 6 is

365

Rules for finding Probability

Suppose the manufacturer specifications of the length of a certain type of computer cable are 2000 ± 10 millimeters. In this industry, it is known that small cable is just as likely to be defective (not meeting specifications) as large cable. That is, the probability of randomly producing a cable with length exceeding 2010 millimeters is equal to the probability of producing a cable with length smaller than 1990 millimeters. The probability that the production procedure meets specifications is known to be 0.99.(a) What is the probability that a cable selected randomly is too large?(b) What is the probability that a randomly selected cable is larger than 1990millimeters?Solution: Let M be the event that a cable meets specifications. Let S and L be the events that the cable is too small and too large, respectively. Then