Chapter 8:Introduction to Probability

• Probability measures the likelihood, or the chance, or the degree of certainty that some event will happen.

• The study of probability provides us with tools for measuring and analyzing uncertainties associated with the occurrence of future events. It allows us to make numerical statements about how likely or unlikely some future event is to occur.

• Probability ranges from 0 to 1, with events closer to 0 being very unlikely to occur, and events closer to 1 being almost likely to occur.

Terminology

• An experiment is any phenomenon whose occurrence yields an observable but unpredictable result.– Examples: tossing a coin and recording what face

turned upthrowing a six sided die and observing what

number was rolled• Sample space refers to the set of all possible

outcomes of an experiment.– Examples: tossing a coin: S= {head, tail}

rolling a die: S= {1,2,3,4,5,6}

Law of Large Numbers

• The law of large numbers states that if we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single value.

• Think about flipping a coin. You might flip a coin 5 times and get a tail 4 out of those 5 times. That doesn’t mean the probability of getting a tail when you flip a coin is 0.50. But as your sample size increases, the probability will approach 0.50.

Classical (Theoretical) Probability• In theoretical probability, an assumption is made

that the outcomes are equally likely and mutually exclusive, meaning that no two events can occur together (for example flipping both a head and a tail on the same flip).

• The probability of event A occurring is defined by the formula

Example: A six-sided die is rolled once. What is the probability that a 5 is rolled?

Empirical Probability

• In empirical probability, outcomes are not equally likely, and thus their weight needs to be accounted for.

• The calculation of empirical probability is based on the assumption that the proportion of times an event actually occurred in the past reflects a blueprint that will be operative in the future.

Example: Dale has a closet with 4 white shirts, 6 blue shirts, and 5 gray shirts. If he randomly selects a shirt from a dark closet, what is the probability that the shirt is white?

Example: Brennan has a jar filled with skittles. There are 10 red skittles, 6 purple skittles, and 8 green skittles. If he randomly reaches into the jar, what is the probability that he draws a green skittle?

Mutually Exclusive Events

Example: Find the probability of randomly selecting the following student(s). a) P(junior)

b) P(sophomore and senior)

c) P(sophomore or senior)

Example: A six-sided die is rolled once. Find the probability of each event occurring.a) P(6)

b) P(4 or 5)

c) P(even number)

d) P(2 and 6)

Non-Mutually Exclusive Events• Two events are non-mutually exclusive if they

could (but do not have to) occur simultaneously.

• The common area where A and B occur is called their intersection.

The General Addition Rule

• The notation “A or B” denotes the event that “either A or B or both occur.”

• To find P(A or B) we add P(A)+P(B) and then subtract the probability that both events occur at the same time (overlap).

• P(A or B) is referred to as the “union.”

Example: 100 workers were sampled to find if they favor or oppose a certain company issue. If one worker is selected at random, find the following probabilities.a) P(blue-collar)b) P(white-collar)c) P(favored the issue)

0.600.400.64

Note: This type of table is known as a two-way table, or a contingency table.

• Are the events “favor” and “blue collar” mutually exclusive?

• In order for this to occur, there has to be no common elements, meaning

• However, we know that

• Therefore, the events are not mutually exclusive.

Above is the results of 81 regular season home baseball games and whether or not fans received a free taco. If a game is randomly selected, find the probability that:

Complementary Events• The probability that an event does not occur is 1

minus the probability that the event does occur.• For example, if event A occurs 70% of the time, it

will fail to occur 30% of the time.• We refer to event “not A” as the complement of

A.

Example:

Example: Two coins are tossed. What is the probability that “two heads do not show?”

S={HH, HT, TH, TT}

Independent vs Dependent Events

• Events A and B are independent if the occurrence of either one of them does not affect the probability of occurrence of the other.

• The result of the second event does is not affected by the result of the first event.

• For example, if a coin is tossed twice in succession, the probability of obtaining a head on the second toss is not in any way influenced by the outcome of the first toss.

In 2010, Jose Bautista of the Toronto Blue Jays led the Major Leagues with 54 home runs. The table above summarizes the outcomes of the Blue Jays’ 81 home games and whether or not Bautista hit at least 1 home run in the game. Question: Are the events “Bautista hits a homerun” and “Blue Jays win” independent?

The events are not independent. Clearly the Blue Jays were more likely to win when Bautista hit a home run.

Dependent Events

• Two events A and B are dependent on each other if the occurrence of one does affect the probability of occurrence of the other.

• As we saw on the previous slide, Bautista hitting a home run and the Blue Jays winning the game were dependent events, as the Blue Jays were clearly more likely to win the game if Bautista homered as opposed to if he did not.

• Conditional probability describes the probability that an event occurs, given that we know that a different event has already occurred.

• Let’s revisit our taco example. If we randomly select one of the victories, what is the probability that a fan at that game received a free taco?

To find this probability, we only care about games in which the team won. A win must occur first. Given the win occurred, what is the probability the fans got a free taco. It may help to eliminate the loss column since we don’t care about games in which the team lost.

• You try this one. Find the probability that the team won the game, given that free tacos were given away.

Expected Value• A random variable takes on numerical values

that describe the outcomes of a chance process.

• In general, for a random variable X, the mean value of X (also called the expected value of X) can be found by multiplying each value of X by its probability and then adding together the products.

Example: The table above uses the results of the World Series from 1945 to 2010 to estimate the probability distribution of X. This probability distribution lists the possible number of games and how often those values occurred. On average, how many games does a World Series last? In other words, what is the mean of the random variable X?

Ex. 2: Hole #13 at the Augusta National golf course is one of the most famous holes in golf. Lined with the course’s signature azaleas (not Iggy), this hole is also a favorite of players for its relative ease. The hole is a par 5, meaning that professional golfers would be expected to complete the hole in 5 strokes. Let X = the score on hole #13 for a randomly selected golfer on day 1 of the 2011 Masters. Above is the probability distribution of X. Calculate the expected value of X.