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
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.
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 𝑷 =
# 𝒐𝒇 𝒇𝒂𝒗𝒐𝒓𝒂𝒃𝒍𝒆 𝒐𝒖𝒕𝒄𝒐𝒎𝒆𝒔
.
𝑻𝒐𝒕𝒂𝒍 # 𝒐𝒇 𝒑𝒐𝒔𝒔𝒊𝒃𝒍𝒆 𝒐𝒖𝒕𝒄𝒐𝒎𝒆𝒔
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 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, …}
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.”
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 Rules
1) 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 3
For 2 six-sided dice being rolled, P(sum=5) =
1. Calculate P(sum≠5).
2. Calculate P(sum<10).
𝟒
𝟑𝟔
≈ 𝟎. 𝟏𝟏𝟏𝟏.
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.
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)
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).
Example 4 Continued
Suppose 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)