Download Probability General Addition Rule: Multiplication Rule for

Probability
General Addition Rule:
Multiplication Rule for independent events:
𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∗ 𝑃(𝐵)
Conditional Probability:
𝑃(𝐵|𝐴) =
𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑃(𝐴)
Two events are independent if:
𝑃(𝐵|𝐴) = 𝑃(𝐵)
Events are mutually exclusive if they have no outcomes in common.
1. At Kennett High School, 5% of athletes play both football and some other contact sport, 30% play football, and 40% play
other contact sports. If there are 200 athletes, how many play neither football nor any other contact sport?
A. 20
B. 70
C. 80
D. 100
E. 130
2. According to a recent national survey of college students, 55% admitted to having cheated at some time during the last year.
What is the probability that for two randomly selected college students, one or the other would have cheated during the past
year?
A. 0.55
B. 0.7975
C. 0.3025
D. 0.2475
E. 0.2025
3. Given two events, A and B, if P(A) = 0.37, P(B) = 0.41, and the P(A or B) = 0.75, then the two events are
A. independent but not mutually exclusive
B. mutually exclusive but not independent
C. mutually exclusive and independent
D. neither mutually exclusive nor independent
E. It cannot be determined from the given information if the two events are independent or mutually exclusive.
4. Security procedures at the U.S. Capitol require that all bags – meaning briefcases, backpacks, shopping bags, any carrying
bag, and purses – must be screened. Currently it is reported that 95% of all bags that contain illegal items trigger the alarm.
12% of the bags that do not contain illegal items trigger the alarm. 12% of the bags that do not contain illegal items trigger the
alarm. If 3 out of every 1,000 bags entering the Capitol contain an illegal item, what is the probability that a bag that triggers
the alarm will contain an illegal item?
A. 0.0233
B. 0.0029
C. 0.9500
D. 0.1140
E. 0.1225
5. Suppose your teacher’s stash of calculators contains 3 defective calculator and 17 good calculators. You select two
calculators from the box for you and your friend to use on the AP Statistics exam. What calculations would you use to
determine the probability that one of the calculators drawn will be defective?
Random Variables
Discrete Random Variable: can take on a finite number of distinct outcomes
Continuous Random Variables: can take on any numeric value within a range of values
Expected Value: 𝜇 = 𝐸(𝑋) = ∑ 𝑥𝑃(𝑋)
Variance: Var(𝑋) = 𝜎 2 = ∑(𝑥 − 𝜇)2 𝑃(𝑥)
Standard Deviation: 𝑆𝐷(𝑋) = 𝜎 = √𝑉𝐴𝑅(𝑋)
Example: At a carnival, a game of chance involves spinning a wheel that is divided into 60 equal sectors. The sectors
are marked as follows: 1 sector: $20, 2 sectors: $10, 3 sectors: $5, 54 sectors: No prize
X
P(x)
$0
54/60
54
$5
3/60
3
$10
2/60
2
$20
1/60
1
𝜇 = 𝐸(𝑋) = 0 (60) + 5 (60) + 10 (60) + 20 (60) = 0.92
54
3
2
1
𝑆𝐷(𝑋) = √(0 − .92)2 (60) + (5 − .92)2 (60) + (10 − .92)2 (60) + (20 − .92)2 (60) = $3.23
Operations on random variables:
If X is a random variable and a and b are fixed numbers, then 𝜇𝑎+𝑏𝑥 = 𝑎 + 𝑏𝜇𝑥
(multiply or add the same fixed number to every outcome)
If X and Y are random variables, then 𝜇𝑋+𝑌 = 𝜇𝑋 + 𝜇𝑌
If X and Y are independent random variables, then 𝜎 2𝑋±𝑌 = 𝜎 2𝑋 ± 𝜎 2 𝑌
6. A radio station is running a lottery to raise money for a local charity. The prizes are $10, $50, $100, and a grand prize
of $1000. The chances of winning these amounts are 0.25, 0.15, 0.09, and 0.01 respectively. What are your total
expected winnings (minus costs) if you pay $1 for a ticket?
A. $29
B. $10
C. 90
D. $290
E. $28
7. The scores for the top three golfers on a high school golf team are used to determine which high schools advance to
the regional level. The Central High team’s top three players have mean scores and standard deviations of:
𝜇𝑥
𝜎𝑥
Player 1
89.5
2.3
Player 2
94.4
4.5
A. 𝜇𝑇 = 281.1, 𝜎𝑇 = 6.38
B. 𝜇𝑇 = 93.7, 𝜎𝑇 = 6.38
D. 𝜇𝑇 = 281.1, 𝜎𝑇 = 3.57
E. 𝜇𝑇 = 281.1, 𝜎𝑇 = 10.7
Player 3
97.2
3.9
C. 𝜇𝑇 = 93.7, 𝜎𝑇 = 3.57
8. Robin owns a bookstore. She is working on a presentation to convince her partner to spend $500 on a catchy
window display. Robin has data to support the fact that if people come in to browse, 62% will make a purchase. Given
that the average purchase is $12.38, what is the expected amount of sales from the next 20 customers who enter the
store?
A. $7.68
B. $ 153.51
C. $247.60
D. $94.60
E. $58.34
Binomial and Geometric Distributions:
𝑛
Binomial Formula: ( ) 𝑝𝑘 𝑞𝑛−𝑘
𝑘
Binomial Distribution
1. Each observation is either success or failure
2. There are a fixed number of observations (n)
3. The observations are independent
4. The probability of success, p, is the same for each observation
Binomialpdf: P(X=k)
Binomialcdf: P(X < or > k)
Expected Value (mean): 𝜇 = 𝑛𝑝
Geometric Distribution
1. Each observation is either success or failure
2. The variable of interest is the number of trials until the first success.
3. The observations are independent
4. The probability of success, p, is the same for each observation.
Geometricpdf: P(X=k)
Gemetric cdf: P(X<k)
1
Expected value (mean): 𝜇 = 𝑝 SD: 𝜎 =
SD: 𝜎 = √𝑛𝑝𝑞
√1−𝑝
𝑝
9. Based on his past performance, the probability that Ben will make a free throw is 0.6. What is the probability that he will
make 3 out of his next 5 free throws?
A. 0.6630
B. 0.0960
C. 0.3456
D. 0.9360
E. 0.01536
10. Based on his past performance, the probability that Ben will make a free throw is 0.6. What is the probability that he will
miss his first three free throws, and then make his fourth one?
A. 0.9744
B. 0.1536
C. 0.8704
D. 0.096
E. 0.0384
11. The Correcto Publishing Company claims that its publications will have errors only twice in every 100 pages. What is the
approximate probability that Anne will read 235 pages of a 790 page book published by Correcto before finding an error?
A. 0.02%
B. 2%
C. 5%
D. 16%
E. 30%
12. According to a CBS/New York Times poll taken in 1992, 15% of the public have responded to a telephone call-in poll. In a
random group of five people, what is the probability that exactly two have responded to a call-in-poll?
A. .138
B. .165
C. .300
D. .835
E. .973
13. A baseball recruiter visits a high school where a player has a batting average of 0.450. What is the probability that the
recruiter won’t see the player get a hit until his third at bat?
A. (0.450)2 (0.55)
B. (0.550)2 (0.450)
3
3
3
C. ( ) (0.450)(0.55)2 D. ( ) (0.550)(0.45)2 E. ( ) (0.450)(0.55)2
1
1
2
Linear Regression
Correlation (r): measures the strength of the linear relationship. Can be between -1 and 1, the closer to -1, and 1 the stronger
the relationship. The sign indicates the direction, positive up, negative down.
Coefficient of determination (𝑟 2 ): Percent of variation in y explained by the linear model with x.
Regression equation: 𝑦̂ = 𝑎 + 𝑏𝑥. a is the intercept and b is the slope.
a: value of y when x is 0
b: for every 1 increase in x, y changes by the slope
Residuals: Difference in the observed and predicted values ( 𝑦̂ − 𝑦)
Residual Plot: should be scattered, small, balanced and have no patterns.
Transforming data:
Power model: 𝑦̂ = 𝑎𝑥 𝑏 (log x, logy)
Exponential model: 𝑦̂ = 𝑎𝑏 𝑥 (x, logy)
19. A scatterplot shows a linear association, and a residual plot for the linear regression shows no pattern. The regression
yielded the following.
Regression Statistics
Coefficients
Multiple R
0.967
Intercept
-1.234
R Square
0.935
Explanatory Var.
2.701
Adjusted R square
0.929
Standard Error
27.877
Observations
14
Which of the following is false?
A. The LSRL is a good linear model for this data
B. The high R value means that it is reasonable to assume a cause and effect relationship between the two variables.
C. Because a new LSRL after removal of one of the points is y = 16.72 + 2.15x, the point that was removed can be
considered an influential point.
D. For every unit increase in x, the y value will increase by approximately 2.701 units.
E. The association is strong and positive.
20. Which of the following statements is (are) correct?
I. Correlation makes no distinction between explanatory and response variables.
II. The sing of r reflects the strength of the association.
III. r measures the strength of a linear relationship only.
A. I only
B. II only
C. III only
D. I and II
E. I and III
22. A regression equation is given as 𝑙𝑜𝑔𝑦̂ = 0.214 − 1.28𝑥. What is the predicted value for y when x =2?
A. -2.346
B. -0.171
C. 0.005
D. 0.167
E. Cannot be determined
Experimental Design and Surveys
23. If you wanted to find the average GPA for seniors at your school who have been accepted into college, what would be the
most appropriate technique to use to gather the data?
A. Census
B. Simple random sample
C. Stratified random sample
D. Systematic random sample
E. Controlled Experiment
25.Two students went to their local shopping mall to conduct a survey. They wanted to know how the local population felt
about boys coloring their hair. Both students had neat haircuts but one had dyed hair and one did not. What type of bias could
occur in the survey?
A. Undercoverage
B. Nonresponse bias
C. Response bias
D. None of the above E. A,B, and C.
26. A study was conducted to determine the benefit of an over the counter medication in reducing the development of disease.
Subjects selected were chosen because they were known to be in high risk group for the disease. The results of the study are
A. not replicable
B. applicable only to the subjects in the study C. not readily generalizable
D. false and misleading
E. valid for all test takers of this over the counter medication.
27 .A study randomly assigned patients to treatment groups to determine the effect of taking aspirin in preventing the
development of colon polyps. One group took an aspirin daily and the other group took a placebo. Neither the patients nor
the doctors knew who was getting which pill. This study is best described as a
A. block design with random assignment.
B. double blind comparative experiment
C. blinded block design observational study
D. blind experiment with random assignment
E. randomly assigned observational study.
28. Which of the following is a true statement about experimental design?
A. Replication is a key component in experimental design. Thus, an experiment needs to be conducted on repeated
samples before generalizing results.
B. Control is a key component in experimental design. Thus, a control group that receives a placebo is a requirement
for experimentation.
C. Randomization is a key component in experimental design. Randomization is used to reduce bias.
D. Blocking eliminates the effects of all lurking variables.
E. the placebo effect is a concern for all experiments.
Probability:
Random Variables
6e
7a
8b
Binomial and Geometric ceaa
9c
10 e
11 a
12 a
Estimation
14. c
15.
16. e
17. a
18. b
Linear Regression
19 d
20 e
21 e
22 c
Experimental Design and Surveys abe
23 a
24 b
25 e
26 c
27 b
28 c
29 e