Probability and RandomnessDay 1

Randomness and Probability

The mathematics of chance is called ___________________.

The probability of any outcome of a chance process is a number between _____________ that describes the proportion of times the outcome would occur in a very long series of ____________.

Short Term and Long Term Behavior

The fact that the proportion of heads in many tosses of a fair coin eventually closes in on 0.5 is guaranteed by the ___________________________.

In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the _______________________a large number of times. According to the law, the ____________ of the results obtained from a large number of trials should be close to the _____________________, and will tend to become closer as _____________________are performed.

The Idea of Probability

20 trails versus 500 trials... See a difference?

Probability Models

In Section 6.1, we used simulation to imitate chance behavior. Fortunately, we don’t have to always rely on simulations to determine the probability of a particular outcome.

Descriptions of chance behavior contain two parts:

The _____________________of a chance process is the set of all

possible outcomes.

A _____________________ is a description of some chance

process that consists of two parts:

• a sample space S and

• a probability for each outcome.

Probability Vocab

• Sample Space: (S)

All the possible outcomes of an event.

• Event:

An outcome or a set of outcomes from a sample space (a subset)

• Complement of any event A:

the event that A does not occur.

Written at AC

Roll The Dice

• List the sample space for rolling a pair of dice

• Suppose “roll a 5” is an event. That event would contain the subset:

Probability Vocab

• Two events are said to be____________________ if and only if the outcome of one has no effect on the outcome of the other.

• Are the tosses of a coin independent event?

• Are the spins on a spinner?

• Is drawing two cards from a deck?

Probability Rules

• The Probability P(A) of an event A satisfies

• If S is the sample space in a probability model then

• Example: When flipping two coins, find the probability of getting at least on head.

• Complement Rule:

Mutually Exclusive

• Events are Mutually Exclusive or Disjoint if:

• they ______ ______ __________ intersect. (They can’t happen at the same time.)

• In Venn diagram form:

Mutually Inclusive

•Events are Mutually Inclusive if:

•they have ______________ outcomes

•In diagram form:

Are they Mutually Inclusive or Mutually Exclusive?

• Graduate from a college with a bachelors degree and a degree in Mathematics.

• In the election, a vote for Trump and a vote for Clinton.

• Catching a trout and catching a fish over 12 inches long.

The Addition Rule (Union)

•Symbol:

•The probability that events A and B occur P(A or B) is,

•If the events are mutually exclusive the

•OR means ____________ the Probabilities!!!

Examples

• 1. You select a card from a deck, what is the probability it is a five or a spade?

• You select a card from a deck, what is the probability it is a spade or a heart?

• You roll a die, what is the probability of getting a six or a prime number?

•Let event A be selecting a letter from HEART and event B be selecting a letter from BEAR.

What is P(Heart U Bear)?

•Let event A be selecting letter of the alphabet from CAT and event B be selecting a letter from HORSE.

What is P(Cat U Horse)?

The Multiplication Rule (Intersection)

• Symbol:

• If events A and B are INDEPENDENT:

• Rule: P(A and B) =

• AND means to _______________ the probabilities!!

The Multiplication Rule (Intersection)• Events A and B are _________________ if:

• One event does change the probability of the next event.

• The Rule:

• P(A and B)=

• In other words:

Important Distinction:

• Disjoint or mutually exclusive ______________be independent!!

• Since they have ______________outcomes, knowing that one occurs means ______________________________

• They are ______________________

Examples

• A general can plan a campaign to fight one major battle or 3 small battles. He believes he has a probability of .6 or winning the large battle and a probability of .8 of winning each small battle. Victories or defeats in small battles are independent. The general must win either the large battle or all three of the small battles. Which strategy should he choose?

Tree Diagrams

• A way to model chance behavior that involves a sequence of outcomes

• Benefits:

•

Example:

Find all the possible outcomes of flipping a coin three times.

For example:

Example: The two –way table below shows the gender and handedness of the students in an AP Stat class.

Gender

Handedness Female Male Total

Left 3 1

Right 18 6

Total

Suppose we choose two students at random.

a) Draw a tree diagram that shows the sample space for

this chance process.

b) Find the probability that both students are left-handed.