Aim: Theoretical Probability Course: Alg. 2 & Trig. Aim: What is the probability of understanding probability? Do Now: How many different possibilities,

Aim: Theoretical Probability Course: Alg. 2 & Trig.

Aim: What is the probability of understanding probability?

Do Now:

How many different possibilities, or outcomes are there with the roll of 1 die?

How many cards are there is a deck of playing cards? How many aces?


Experimental or Empirical Probability

Probability - The likelihood an event will occur.

gathering data through observations to determine the likelihood of an event occurring is called experimental or empirical probability and is expressed mathematically as a ratio.

Experimental Probability =

number of times event occurs

number of trials

Ex. Eli Manning completes 30 of 40 passes.

The probability of a completion is ¾.


Theoretical Probability

Theoretical probabilityTheoretical probability - the number of ways that an event CAN occur divided by the total number of possible outcomes.

outcome - a result of some activitysample space - all possible outcomes event - any subset of the sample space

P En En S

( )( )( )

P(E) - probability of event E n(E) - the number of ways event E can occur n(S) - the number of possible outcomes

Probability is expressed as a ratio or a percent.

Ex. The probability of rolling a pair of dice a getting a twelve is 1 in 36 or you have less than a 3% chance of rolling a 12.


Probability

Sample space: n outcomes

Event A:m outcome

( )( )

( )

n AP A

n S

In a class of 147 students, 95 are taking math (M), 73 are taking science (S) and 52 are taking both.

M S43 52 21

P(taking M or S or both)

31


Geometric Probability

area of center circle(20)

area of entire dart boardP

2010521

What must we know to find the probability of scoring a 20 on the toss of a dart?

5r

The areas of the circle and each of the rings

If the radius of the circle is 5r and each ring is r wide, what is the probability of scoring 20 on one toss?

2

2

1

255

r

r


Numerical Values of Probability

• The values of probability ratios range from 0 through 1.

• A probability of 0 means that the outcome will never happen.

• A probability of 1 means that the outcome will be certain to happen.

• The closer a probability is to 0, the less likely the outcome.

• The probability of event (A) plus the probability of "not A” or ~A, equals 1: P(A) + P(~A) = 1

Values for a Probability Ratio


Numerical Values of Probability

•The probability of an event E must be equal to or greater than zero (0) and less than or equal to one (1).

Values for a Probability Ratio

0 1.5AlwaysEven

ChanceNever Not

Likely Likely

.75.25

What is the probability of rolling 13 with a pair of dice? 0


The Spinner

P En En S

( )( )( )

What is the probability of landing on red?


What’s the difference between spinners?

Uniform Biased or weighted

Each event has an equal chance of

occurring

Each event has an unequal chance of

occurring

P(Red) = 1/8 P(Red) = 2/8

P(Blue) = 1/8 P(Blue) = 2/8


Model Problem

A letter is chosen at random from the word REED.

A. List the sample space.

B. Find the probability of choosing an E.

A. {R, E, E, D}

P(E) = Number of E’sNumber of letters

P(E) = 24

or 1/2

B.


Model Problems

A bag contains a nickel, a dime and a quarter. A person selects one of the coins. What is the probability that the coin is worth

A. Exactly ten cents

B. Exactly three cents

C. More than three cents.

Main Idea

P(coin is worth .10) = 1/3

P En En S

( )( )( )

P(coin is worth .03) = 0/3

P(coin is worth >.03) = 3/3 = 1


Model Problems

The name of a quadrilateral is selected at random from the set {parallelogram, rhombus, rectangle, square, trapezoid, isosceles trapezoid}. What is the probability of selecting the name of a quadrilateral that has:

a. Both pairs of opposite sides congruent?

b. Congruent diagonals.

c. Perpendicular diagonals

Subset: {parallelogram, rectangle, square, rhombus}

P = 4/6

Subset: {rectangle, square, isosceles trapezoid}

P = 3/6

Subset: {square, rhombus} P = 2/6


Model Problems

1. An urn (jar) contains 4 red marbles and 6 white marbles. If one marble is drawn at random, what is the probability that it will be blue?

2. The probability that we will win the baseball game is x/5. What is the probability that we will not win the game?

3. If t represents the probability the event T will occur, what is the probability that event ~T will not occur?


Model Problem

In the Sullivan family, there are two more girls than boys. At random, Mrs. Sullivan asks one of her children to go to the store. If she is equally likely to ask any one of her children, and the probability that she asks a girl is 2/3, how many boys and how many girls are there is the Sullivan family?

x = number of boys

x + 2 = number of girls

2x + 2 = number of children

P(girl) = Number of girlsNumber of children

x + 22x + 2

23

= 4x + 4 = 3x + 6

x = 2

2

4


Model Problem

In a class of 147 students, 95 are taking math (M), 73 are taking science (S) and 52 are taking both.

M S43 52 21

P(taking M or S or both)

of M or S or both(M or S or both)

Total number of students

nP

31

Documents

Aim: Theoretical Probability Course: Alg. 2 & Trig. Aim: What is the probability of understanding probability? Do Now: How many different possibilities,