Section 5 1 Discrete Probability Probability Distributions x

Section 5. 1 Discrete Probability

Probability Distributions • x 1 P(x) 0. 4 2 3 4 5 x 4 6 8 0. 3 0. 1 0. 3 1. 4 P(x) 1/4 0 10 12 1/8 3/8

Discrete vs. Continuous • Discrete – can be counted, whole numbers • Continuous – cannot be counted, fractions, decimals

Expected Value • x 4 5 6 7 8 P(x) 0. 2 0. 1 0. 05 0. 45

Variance and Standard Deviation • x 2 P(x) 0. 5 3 4 0. 05 x P(x) 2 0. 5 -1. 65 2. 7225 1. 36125 3 0. 05 -0. 65 0. 4225 0. 021125 4 0. 05 . 35 0. 1225 0. 006125 5 0. 1 1. 35 1. 8225 0. 18225 6 0. 3 2. 35 5. 5225 1. 65675 5 6 0. 1 0. 3 Sum = 3. 2275 = Variance

Profit and Loss w/ Probability •

Example • If you draw a card with a value of 2 or less from a standard deck of cards, I will pay you $303. If not, you pay me $23. (Aces are the highest card in the deck) • Find the expected value of the proposition.

Solution •

Example (part 2) • If you played the same game 948 times, how much would you expect to win or lose?

Solution (part 2) •

Creating Probability Distribution w/ Tree Diagram • The number of tails in 4 tosses of a coin. x P(x) 0 1 2 3 4

Pascal’s Triangle 2 tosses 3 tosses 4 tosses

Example Make sure you simplify all fractions. To get the total you can add the numbers in the row or take 2 to the power of the number of times you are choosing or flipping.

Finding Probability A =. 5 B =. 9 C =. 2 D =. 5

Section 6 -1 Introduction to Normal Curve

Normal Curve

Example

Section 6 -2 Finding area under the Normal Curve

Area Under a Normal Curve • Using z-scores (standard scores) we can find the area under the curve or the probability that a score falls below, above, or between two values. • The area under the curve is 1. • The mean (or z=0) is the halfway point, or has an area of. 5000. • Values are listed to four decimal places.

To How the Area under the Curve • If asked for the area to the left, find the value in the chart. • If asked for the area to the right, find the value and subtract from 1. Alternate Method: Find the opposite z-score and use that value. • If asked for the area between two z-scores, find the values and subtract. • If asked for the area to the right and to the left of two numbers, find the values and add.

1 - z-score Alternate Method

Examples • Find the area: – To the left of z=2. 45 – To the right of z=2. 45 – Between z=-1. 5 and z=1. 65 – To the left of z=1. 55 and to the right of z=2. 65 – To the left of z=-2. 13 and to the right of z=2. 13

Solutions • • • . 9929. 0071. 9960 -. 0668=. 9292. 0606+. 0013=. 0619. 0166+. 0166=. 0332

Problems with greater than and less than • Some problems will have greater than or less than symbols. • P(z<1. 5) is the same as to the left of z=1. 5 • P(z>-2. 3) is the same as to the right of z=-2. 3 • P(-1. 24<z<1. 05) is the same as between z=1. 24 and z=1. 05 • P(z<1. 02 and z>. 02) is the same as to the left of z=1. 02 and to the right of z=. 02

Section 6 -3 Finding area after finding the z-score

How to solve • Find the z-score with the given information • Determine if the value is to the left, right, between, or to the left and right. • Look up values in the chart and use directions from 6 -2.

Examples

Solutions • • P(0<z<1. 5) =. 4332 P(z<0) =. 5000 P(z>2) =. 0228 P(-. 75<z<0. 5) =. 4649

Section 6 -4 Finding Z and X

Finding Z • If the value is to the left: – Find the probability in the chart and the z-score that corresponds with it. • If the value is to the right: – Subtract the value from one, find the probability and the z-score that corresponds with it. OR – Find the value and the corresponding z-score and change the sign.

Finding Z • If the value is between: – Divide the area by 2, then add. 5, then find the corresponding z-score. OR – Subtract the area from 1, divide by two, then find the corresponding z-score. • If the value is to the right and left: – Divide the area by 2, then find the corresponding z -score.

Examples • Find the z-score that corresponds with: – Area of. 1292 to the left – Area of. 3594 to the right – Area of. 7154 between – Area of. 8180 to the left and the right

Solutions • • -1. 13. 36 -1. 07 and 1. 07 -. 23 and. 23

Word Problems •

Word Problems