Chapter 6Lesson 6.2

Probability6.2: Definition of Probability

What is Probability?

Two different approaches to probability

The Classical Approach

When the outcomes in a sample space are equally likely, the probability of an event E, denoted by P(E), is the ratio of the number of outcomes favorable to E to the total number of outcomes in the sample space.

outcomes totaloutcomes favorable

)( EP

Examples: flipping a coin, rolling a die, etc.

On some football teams, the honor of calling the toss at the beginning of the football game is determined by random selection.Suppose this week a member of the 11-player offensive team will be selected to call the toss. There are five interior linemen on the offensive team.If event L is defined as the event that an interior linemen is selected to call the toss, what is probability of L?

P(L) = 5/11 = .4545

Consider an archer shooting arrows at a target.

The probability of getting a bulls’ eye should be the ratio of the area of the inner circle to the area of the entire target.

What if a very experienced archer were shooting the arrows? Would the probability of a bull’s eye still be the same?The classical approach doesn’t

work for every situation.

The Relative Frequency ApproachThe probability of event E, denoted by P(E), is defined to be the value approached by the relative frequency of occurrence of E in a very long series of trials of a chance experiment. Thus, if the number of trials is quite large,

trials of numberoccurs E times of number

)( EP

Pick a Card…Any Card

• What’s the probability of drawing diamonds from a standard deck?

What’s the P(diamond)?

Trial # # of outcomes # of s % s

# of Outcomes

% d

iam

onds

Consider flipping a coin and recording the relative frequency of heads.

When the number of coin flips is small, there is a lot of variability in the relative frequency of “heads” (as shown in this graph). What do you notice in the graph at the right?

Consider flipping a coin and recording the relative frequency of heads.

The graph at the right shows the relative frequency when the coin is flipped a large number of times. What do you notice in this graph at the right?

Law of Large Numbers

As the number of repetitions of a chance experiment increase, the chance that the relative frequency of occurrence for an event will differ from the true probability by more than any small number approaches 0.

OR in other words, after a large number of trials, the relative

frequency approaches the true probability.

Notice how the relative frequency of heads approaches ½ the

larger the number of trials!

Homework

• Reading Notes 6.3