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11.2 Probability
number of times the event occurs
number of trials
Experimental probability: P(event)=
*this is based on what ACTUALLY happens when YOU do the experiment
Theoretical probability: the sample space that has equally likely outcomes
*this is based on what SHOULD happen (theoretically)
Simulation: a model of the event
Sample space: the set of all possible outcomes to an experiment
Examples:
Find the experimental probability of a quarterback completing his next pass if he has:
1) Completed 30 of his last 40 passes
2) Completed 36 of his last 45 passes
Find the theoretical probability of each event when rolling a standard number cube.
3) P(3)
4) P(2 or 4)
5) P(not an even number)
11.3-4 Probability of Multiple Events
dependent events: when the outcome of one event effects the outcome of another
independent events: outcomes not dependent on other events
mutually exclusive events: events that can NOT happen at the same time
11.4 Conditional Probability
Conditional probability: the probability than an event B will occur given that event A has already happened *events
must be dependent* (Indicated with | sign)
Tree diagram: a way of organizing events by showing all possible outcomes on a diagram
Examples:
Tell whether outcomes of each trial are dependent or independent events:
1) A month is selected at random; a number from 1 to 30 is selected at random
2) A letter of the alphabet is selected at random; a remaining letter is selected at random.
Q and R are independent events. Find P(Q and R)
3) P(Q)= 1/4 , P(R)= 2/3
4) P(Q)= 0.6, P(R)= 0.9
Two number cubes are rolled. State whether the events are mutually exclusive.
5) The numbers are equal; the sum is odd
6) It’s a multiple of 4, the numbers are different
S and T are mutually exclusive. Find P(S or T).
7) P(S)= 5/8, P(T)= 1/8
8) P(S)= 12%, P(T)=27%
A standard number cube is tossed. Find each probability.
9) P(3 or odd)
10) P(even or less than 4)
Use the table to find each probability.
11) P(has diploma and experience)
12) P(no diploma | has experience)
13) P(bachelors)
14) P(not an associates, given the recipient is male)
15) Never had a pet, and has no pet now
16) There’s a 60% chance of a light snow and 40% chance for heavy snow today. When there is light snow, the
school remains open 70% of the time; heavy snow the school remains open 20% of the time. Draw a tree
diagram and find the probability of the school closing today.
11.6 Analyzing Data
measure of central tendency: indicates the “middle” of the data set (most common measures are mean, median, and
mode)
sum of data values
mean: average or number of data values
median: the middle value when numbers are listed in order
mode: the most frequently occurring values *bimodial data means there are TWO modes*
outlier: a value that is substantially different than the rest of the data
Example:
1) Find the mean median and mode. Discard the outlier. Find the new mean, median and mode. Which measure
of central tendency is most affected?
56 65 73 59 98 65 59
11.8 Samples and Surveys
*You CAN PRINT THESE (it’s a lot of writing)!*
Population- all the members of a set
Sample- part of a population
Bias- a systematic error when a part of a population is overrepresented or underrepresented
Sample methods:
Convenience- select any members of the population who are conveniently and readily available
Self-Selected- select only members of the population who volunteer for the sample
Systematic- order the population in some way and select from it at regular intervals
Random- all members of the population are equally likely to be chosen
Study Methods:
Observational- measure or observe members of a study in such a way they are not affected by the study
Controlled Experiment- divide the sample into two groups; impose a treatment on one group but not the other “control”
group. Then, compare both groups
Survey- ask every member of the sample a set of questions
Examples:
11.7 Standard Deviation
measure of variation: describes how data in a data set are spread out
Variance/standard deviation: shows how much data values deviate from the mean
Σ = capital sigma (sum)
σ = lower case sigma (standard deviation)
σ2 = (variance)
𝑥̅ = mean
Examples: *leave 8 lines for 1-2 and 2 lines for 3-4*
1)
2)
3)
4)
What are the mean, variance, and standard deviation of these values?
52 63 65 77 80 82
6.5 5.8 3.9 5.7 4.2
Determine the whole number of standard deviations from the mean that include all the data values.
Values: 19, 15, 20, 22, 21, 18, 21, 13, 19, 24, 17, 14, 12, 15, 14
Mean: 17.6; Standard Deviation: 3.5
Values: 4, 3, 5, 4, 6, 8, 1, ,3, 2, 5, 6, 4, 7, 5, 3
Mean: 4.4; Standard Deviation: 1.82
11.10 Normal Distributions
discrete probability distribution- has a finite number of possible events or values
continuous probability distribution- can be any value in an interval of real numbers
normal distribution- has data that varies normally from the mean; has a normal curve
*If data is skewed, it does not vary predictably from the mean and you cannot use standard deviation
Examples:
1. You track the number of words in your text messages for a month and sketch the bar graph shown. The
number of words is normally distributed about a mean of 10. About what percentage of your texts are
between 9 and 11 words long?
2. The average score on a test was an 82 and the standard deviation of the normally distributed scores was 5.
Sketch a normal curve showing the scores and three standard deviations from the mean. Label the x-values.
3. Using the normal curve from Example 2, what percent of students scored between 72 and 82 points.
4. The scores on the Algebra 2 final are approximately normally distributed with a mean of 150 and a standard
deviation of 15.
a) What percentage scored above a 180?
b) If 250 students took the test, approximately how many scored above a 135?