MATH 2400Ch. 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 1Player 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.

Example 1 Continued…Player A has a mean batting average of .345 with a standard deviation of .085.

2. For any random game X, calculate P(X<0.25). Draw a shaded bell curve representing this data.

3. For any random game X, calculate P(X>.420). Draw a shaded bell curve representing this data.

Example 1 Continued…Player A has a mean batting average of .345 with a standard deviation of .085.

2. 4. For any random game X, calculate P(0.300<X<0.400). Draw a shaded bell curve representing this data.

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.

Probability ModelsA 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 complexFlipping a coin: {H, T} Car Tags: {111AAA, 111AAB, …}

Example 2List 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.”

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.

Probability Rules1) Any probability is a number between 0 and 1.2) All possible outcomes together must have a probability of 1.3) If two events have no outcomes in common, the probability that

one or the other occurs is the sum of their individual probability.4) 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.

Probability Rules (Mathematical Notation)

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

2. Calculate P(sum<10).

Discrete Probability ModelsA 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.

Continuous Probability ModelsConsider 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)

Example 4Consider 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).

Example 4 ContinuedSuppose a random number X was generated.1) Calculate P(0.3 < X < 0.7)

2) Calculate P(X < 0.5)

3) Calculate P(X > 0.8)

4) Calculate P(X < 0.5 or X > 0.8)

Example 4 Continued

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).

Example 6 (Exercise 10.15)

Example 7 (Exercise 10.17)