Section 6.1 and 6.2 Probability

Probability; it’s all chance!

The big idea in probability is that chance behavior is unpredictable in the short run but follows a regular, predictable pattern in the long run.

Probability is a mathematical model for what SHOULD happen!!

A Common Question

The probability of tossing a coin and it landing on Heads is 0.5. Theoretically, then, if I toss the coin 10 times, I should get 5 Heads. However, with such a small number of tosses there is a lot of room for variability. There are two games involving flipping a coin.

Game 1: You win $2 if you throw 40% - 60% heads.

Game 2: You win $10 if you throw more than 75% heads.

For which game would you rather toss the coin 5 times? 500 times?

The Idea of Probability

Random does NOT mean haphazard. Random events have an order that emerges in the long run.

Random – individual outcomes are uncertain, but there is a regular distribution of outcomes in a large number of repetitions.

Probability – proportion of times the outcomes would occur in a very long series of repetitions.

Independent – the outcome of one trial does not influence the outcomes of any other trial (ex. Rolling a die).

Properties

Remember that probabilities are like proportions; therefore, probabilities will always be a number between 0 and 1 (inclusive).

The probability that I draw a 15 from a standard deck of cards is 0.

The probability that I draw a red or a black card from a standard deck is 1.

Probability Models

Sample space S is the set of all the possible outcomes.

If I toss 2 coins, and count the number of heads, the sample space is {0, 1, 2}.

An event is any outcome or set of outcomes of a random phenomenon.

An event for tossing 2 coins might be getting at least 1 head.

A probability model is a mathematical description – it lists the sample space and the probabilities associated with each event.

Word of the Day: Enumeration

It will be critical for you to be able to list (enumerate) all the possible outcomes of a random phenomenon.

Examples: List all the possible outcomes. How many are there? Then write the sample space.

Roll 2 dice

Roll 1 die and toss a coin

Two techniques can help us with enumeration…

Tree Diagrams Draw the “branches” for the first task, and then from

each of those branches, draw the branches for the second task.

Let’s do this to represent the sample space for throwing one die and flipping a coin.

Multiplication Principle Multiply the number of outcomes for the first task by

the number of outcomes for the second task. Do this for rolling two dice.

Example

List the sample space for how many boys you could have within three children.

Give the probability model for this situation.

Probability Rules

Any probability is a number between (and including) 0 and 1.

All possible outcomes together must have probability = 1.

The probability that an event does NOT occur is one minus the probability that the event does occur.

If two events have no outcomes in common, then probability that one OR the other occurs is the sum of their individual probabilities.

Any probability is a number between (and including) 0 and 1.

In symbols:

0 ≤ P(A) ≤ 1

Probabilities can not be negative!

Probabilities can not be greater than 1!

All possible outcomes together must have probability = 1.

In symbols.. The probability of the whole sample space: P(S) = 1 Sample question: A randomly selected student is asked to respond

yes, no, or maybe to the following question, “Do you intend to vote in the next presidential election?” The sample space is {yes, no, maybe}. Which of the following represents a legitimate assignment of probabilities for this sample space? A) .4, .4, .2 B) .4, .6, .4 C) .3, .3, .3 D) .5, .3, -.2 E) None of the above

The probability that an event does NOT occur, called the complement( )

is one minus the probability that the event does occur.

In symbols:

Sample question: If you choose a card at random from a well-shuffled deck of 52 cards, what is the probability that the card is not a heart?

( ) 1 ( )CP A P A

CA

P(Heart) = 13/52 = ¼

P(Not a Heart) = 1 – P(Heart) = 1 – ¼ = ¾

If two events have no outcomes in common, then probability that one OR the other occurs

is the sum of their individual probabilities.

Having no outcomes in common is called DISJOINT or MUTUALLY EXCLUSIVE.

This rule is very important! It’s called the addition rule for disjoint events.

In symbols:

P(A or B) = P(A U B) = P(A) + P(B)

This is read

“A union B.”

Sample Question

Which of the following pairs of events are disjoint? a) A: The odd numbers; B: the number 5

b) A: the even numbers; B: the numbers greater than 10

c) A: the numbers less than 5; B: all negative numbers

d) A: the numbers above 100; B: all negative numbers

e) A: negative numbers; B: odd numbers

More Symbols

{A ∩ B} “A intersect B” – this means the

outcomes that A and B have in common

If A ∩ B = Ø, then A and B are disjoint!

What is A U AC?

What is A ∩ AC?

WHY? 1

Ø

Another thing to remember

The probability that something will occur AT LEAST ONCE

1 – P(event does not happen)how many attempts

Example: If the probability that I pass any given Statistics test is 70% and I have 10 tests this semester, what is the probability that I pass at LEAST ONE?!

Homework

6.4, 6.9, 6.11, 6.14, 6.18, 6.34, 6.36