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MATH 2400

MATH 2400. Ch. 10 Notes. So…the Normal Distribution. Know the 68%, 95%, 99.7% rule Calculate a z-score Be able to calculate Probabilities of… X < a (X is less than a) X > a (X is greater than a) a < X < b (X is between a and b). Example 1.

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MATH 2400

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  1. MATH 2400 Ch. 10 Notes

  2. So…the Normal Distribution. • Know the 68%, 95%, 99.7% rule • Calculate a z-score • Be able to calculate Probabilities of… • X < a (X is less than a) • X > a (X is greater than a) • a < X < b (X is between a and b)

  3. Example 1 Player A and Player B are both candidates for being drafted by a professional baseball team. Player A has a mean batting average of .345 with a standard deviation of .085. Player B has a mean batting average of .362 with a standard deviation of .119. 1. Which player should be drafted? Fully explain why you think so.

  4. Example 1 Continued… Player A has a mean batting average of .345 with a standard deviation of .085. • For any random game X, calculate P(X<0.25). Draw a shaded bell curve representing this data. • For any random game X, calculate P(X>.420). Draw a shaded bell curve representing this data.

  5. Example 1 Continued… Player A has a mean batting average of .345 with a standard deviation of .085. • 4. For any random game X, calculate P(0.300<X<0.400). Draw a shaded bell curve representing this data.

  6. Ch. 10 (For Real This Time) Probability is simply . This allows us to calculate the theoretical probability of an event happening. Experimental Probability can be calculated by doing many trials of an experiment.

  7. Probability Models A sample space, S, of a random phenomenon is the set of all possible outcomes. An event is an outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space. A probability model is a mathematical description of a random phenomenon consisting of two parts: a sample space S and a way of assigning probabilities to events. Sample spaces can be very simple or very complex Flipping a coin: {H, T} Car Tags: {111AAA, 111AAB, …}

  8. Example 2 List a sample space for the sum of 2 six-sided dice. For kicks and giggles, let’s calculate the probability of getting a sum of “6.”

  9. Example 2 Continued… Fill in the following table with the probabilities of rolling each sum if 2 six-sided dice are rolled. Do the same for the difference of the two dice. Create a table of values and Probability.

  10. Probability Rules 1) Any probability is a number between 0 and 1. • All possible outcomes together must have a probability of 1. • If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probability. • The probability that an event does not occur is 1 minus the probability that the event does occur. (The probability that something does not occur, is called its complement.

  11. Probability Rules (Mathematical Notation)

  12. Example 3 For 2 six-sided dice being rolled, P(sum=5) = . 1. Calculate P(sum≠5). 2. Calculate P(sum<10).

  13. Discrete Probability Models A probability model with a finite sample space is called a discrete probability model. On slide 9, you filled out a table with the sample space listed along with the relative probabilities.

  14. Continuous Probability Models Consider a situation in which we asked a computer to generate a random number (when this is done, it spits out a random number between 0 and 1, like 0.2893511). If we wanted to calculate P(0.3≤X≤0.7) it would be very difficult because there is an infinite interval of possible values. *Also, because X represents the value of a numerical outcome of a random phenomenon, we call it a random variable. For situations like this, we have to use a different model. Areas under a density curve. Any density curve has area exactly 1 underneath it, corresponding to total probability 1. A continuous probability model assigns probabilities as areas under a density curve. (Think back to shading under the bell curve)

  15. Example 4 Consider the situation in which a computer is asked to generate random numbers. 10,000 numbers were generated and a histogram of the data is given. This data represents an uniformally distributed data set (rectangular).

  16. Example 4 Continued Suppose a random number X was generated. • Calculate P(0.3 < X < 0.7) • Calculate P(X < 0.5) • Calculate P(X > 0.8) • Calculate P(X < 0.5 or X > 0.8)

  17. Example 4 Continued

  18. Example 5 (Normal Distribution) Suppose the heights of young women has a mean μ=64.3 inches and standard deviation σ=2.7. Calculate P( 68 < X < 70).

  19. Example 6 (Exercise 10.15)

  20. Example 7 (Exercise 10.17)

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