LAW OF LARGE NUMBERS OBJECTIVE Understand the Law

LAW OF LARGE NUMBERS

OBJECTIVE • Understand the Law of Large Numbers • Calculate basic probabilities.

RELEVANCE To find the likelihood of an event occurring by observing the long-term proportion in which a certain outcome is observed.

Simulation: Proportion of Heads Flip the Coin Flip Simulation 10 Times 5000 Times What Do You Notice? • Short Term (Fewer Flips): The observed proportion of heads is different and unpredictable for each experiment. • Long Term: As the # of flips increases the proportion heads toward 50%.

Did You Notice? • As the number of flips increased, the closer the proportion came to the expected value of 50%. • This is the basic premise of probability. Probability deals with experiments that yield random short-term results or outcomes yet reveal long-term predictability.

Definition • The long-term proportion in which a certain outcome is observed is the probability of that outcome.

In Other Words • Probability describes how likely it is that some event will happen. • If we look at the proportion of times an event has occurred over a long period of time (or over a large number of trials), we can be more certain of the likelihood of its occurrence. • This phenomenon is referred to as the Law of Large Numbers.

Law of Large Numbers

Labs Let’s see the law of large numbers at work.

Terminology Term • Experiment Definition • Any process with uncertain • Sample Space (S) results that can be repeated. • The collection of all possible outcomes of a probability experiment. • Event (E) • Any collection of outcomes from a probability experiment. • Unusual Event • An event that has a low probability of occurring.

Find the Sample Space Event Sample Space Toss 1 Coin H or T Answer a T or F Question Toss 2 Coins T or F HH, TT, HT, TH

A probability experiment consists of rolling a single fair die. Identify the outcomes: Event 1 Event 2 Event 3 Roll a 1 Roll a 2 Roll a 3 Event 4 Event 5 Event 6 Roll a 4 Roll a 5 Roll a 6 Determine the Sample Space: Define the event E = “roll an even number”

A probability experiment consists having 2 children. Identify the outcomes: Event 2 Event 1 Boy, Girl Event 3 Event 4 Girl, Boy Girl, Girl Determine the Sample Space: Define the event E = “have one boy”

Notation • Use of the notation P(E) means “the probability that event E occurs. ”

Probability Rules Rule 1 • Rule 2 • No negative probability No probability bigger than 1.

Probability Rules Rule 3 Example If an event can NEVER happen the probability is 0. What is the probability of rolling a “ 9” on a fair die?

Probability Rules Rule 4 If an event is CERTAIN to happen the probability is 1. Example What is the probability of rolling a “number less than 7 ” on a fair die?

Example If a year is selected at random, find the probability that Thanksgiving Day will be on a Wednesday. •

Example What is the probability that New Year’s will fall on January 1 st? •

3 Probability Methods • Empirical • Experimental • Classical (Theoretical) • Assumes all outcomes in • Subjective • The probability of an S are equally likely. event is estimated by using knowledge of the relevant circumstances.

$How are probabilities expressed? • Probabilities are expressed as reduced fractions, decimals rounded to$

How are probabilities expressed? • Probabilities are expressed as reduced fractions, decimals rounded to 2 or 3 decimal places, or, where appropriate, percentages Examples: 1. 2. 0. 5 3. 50%

Empirical Probability • Probabilities computed in this manner rely on experimental evidence. • In other words, it is evidence based on the outcomes of a probability experiment.

Empirical Probability Example • If a person rolls a die 40 times and 9 of the rolls results in a “ 5”, what empirical probability was observed for the event “ 5”? • Answer:

Example: Empirical Probability • Pass the Pigs is a game in which pigs are used as dice. Points are earned based on how the pigs land. There are 6 possible outcomes. 52 students rolled the pigs 3939 times. The # of outcomes each occurred in shown.

Find the P(Side With Dot) Estimate the probability that a pig lands on a “side with a dot. ”

Classical Probability • The classical method does not require that a probability experiment actually be performed (like the empirical). • It relies on counting techniques. • It requires equally likely outcomes (each has the same probability of occurring).

Classical Probability P(E) Probability of event E S Sample Space N(E) Number of outcomes in E N(S) Number of outcomes in the sample space.

Classical Probability Example What is the probability of drawing a queen from a deck of cards?

Classical Probability Example If a family has 3 children, find the probability that all 3 children are girls. You are going to have to look at the sample space before you can answer this one.

Looking for all 3 girls • Sample Space: BBB BBG BGB GBB GGG GGB GBG BGG • Answer:

Classical Probability Example A card is drawn from an ordinary deck. Find these probabilities: Event Probability

Classical Probability Example • What is the probability of rolling a die and getting a “ 5”?

Empirical vs. Classical Empirical Classical • If a person rolls a die 40 • What is the probability of times and 9 of the rolls results in a “ 5”, what empirical probability was observed for the event “ 5”? rolling a die and getting a “ 5”?

Subjective Probabilities • A subjective probability of an outcome is a probability obtained on the basis of personal judgment. (Has high degree of personal bias) • Ex: If a sports reporter is asked what he thinks the chances are for the Boston Red Sox to play in the World Series, he will make an educated guess. He may say 20%. His probability is not based on repeating the season. It is based on his own personal judgment which is very subjective.

Examples of Subjective Probability • An economist predicts that there is a 20% chance of recession next year. • Ask New York Yankee fans before the start of the baseball season the chances of winning the World Series.

Complement of an Event • Suppose that the probability of an event E is known and we would like to determine the probability that E does NOT occur. • This can be easily accomplished using the idea of complements.

Definition •

Complement Rule •

Example • According to the American Veterinary Medical Association, 31. 6% of American households own a dog. • What is the probability that a randomly selected household will NOT own a dog?