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Section 5.1

Section 5.1. Discrete Probability. Probability Distributions. A probability distribution is a table that consists of outcomes and their probabilities. To be a probability distribution it must have the following properties: Each probability must be The probabilities must have a sum of 1.

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Section 5.1

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  1. Section 5.1 Discrete Probability

  2. Probability Distributions • A probability distribution is a table that consists of outcomes and their probabilities. • To be a probability distribution it must have the following properties: • Each probability must be • The probabilities must have a sum of 1.

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

  4. Expected Value • Expected value is the same as a weighted mean. • Formula: • Expected Value = = 6.35

  5. Variance and Standard Deviation • Variance: where mean is the expected value. • Standard Deviation: square root of the variance

  6. Profit and Loss w/ Probability • To determine the profit or loss using probability you will use the expected value for each event. • Formula: Profit minus loss: • is the value of the profit or what you receive • is the value of the loss or what you pay.

  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 • Find the probability of drawing a card with a value of 2 or less. • Find the value of drawing a card greater than 2. • Determine and . • Fill in formula. • So for each round that is played the is an expected gain of $2.08. • If there is a loss, the value would be negative.

  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) • Take the profit or loss from one round and multiply by the number of times played.

  11. Creating Probability Distribution w/ Tree Diagram • The number of tails in 4 tosses of a coin.

  12. Section 6-1 Introduction to Normal Curve

  13. Normal Curve

  14. Example

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

  16. 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.

  17. 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.

  18. 1 - z-score Alternate Method

  19. 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

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

  21. 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

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

  23. 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.

  24. Examples

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

  26. Section 6-4 Finding Z and X

  27. 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.

  28. 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.

  29. 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

  30. Solutions • -1.13 • .36 • -1.07 and 1.07 • -.23 and .23

  31. Word Problems • Determine if the problem is looking for less than, greater than, between, or less than and greater than. • Find the z-score(s). • Use the formula to solve for x. • Some problems you will have two solutions.

  32. Word Problems

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