1. Section 5.1 Discrete Probability

2. Probability Distributions x4681012 P(x)1/4 01/83/8 x12345 P(x)0.40.30.10.31.4

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

4. Expected Value x45678 P(x)0.2 0.10.050.45

5. Variance and Standard Deviation x23456 P(x)0.50.05 0.10.3 xP(x) 20.5-1.652.72251.36125 30.05-0.650.42250.021125 40.05.350.12250.006125 50.11.351.82250.18225 60.32.355.52251.65675 Sum = 3.2275 = Variance

6. Profit and Loss w/ Probability

7. 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.

8. Solution

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

10. Solution (part 2)

11. Creating Probability Distribution w/ Tree Diagram The number of tails in 4 tosses of a coin. x01234 P(x)

12. Pascal’s Triangle 2 tosses 3 tosses 4 tosses

13. 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.

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

15. Section 6-1 Introduction to Normal Curve

16. Normal Curve

17. Example

21. Section 6-2 Finding area under the Normal Curve

22. 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.

23. 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.

24. 1 - z-score Alternate Method

25. 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

26. Solutions.9929.0071.9960-.0668=.9292.0606+.0013=.0619.0166+.0166=.0332

27. 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.02) is the same as to the left of z=1.02 and to the right of z=.02

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

29. 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.

30. Examples

31. Solutions P(0<z<1.5) =.4332 P(z<0) =.5000 P(z>2) =.0228 P(-.75<z<0.5) =.4649

32. Section 6-4 Finding Z and X

33. 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.

34. 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.

35. 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

36. Solutions -1.13.36 -1.07 and 1.07 -.23 and.23

37. Word Problems