CSE7315c – Fundamentals of Probability and Statistical Methods Activity Sheet
Learning outcomes:
After solving these exercises, you should be able to understand the following: A. B. C. D. E. F.
Probability Conditional probability Mutually exclusive and independent events Expectations Bayes Rule Normal distributions
1. The probability the person is female given that person attends the INSOFE CPEE program is? We have 40 students and 4 of them are Female. 2. The below table summarizes two variables for 28,975 visits to cleartrip.com. The host ho st that sent the visitor and whether a purchase was made.
Host Yahoo
Purchase
Google
shopclues.com
Total
No
268
27,069
656
27,993
Yes
27
926
29
982
295
27,995
685
28,975
Total
a. Convert the above table into the probabilities. To convert the above table into probabilities, think about the next visitor to Cleartrip.com from one of the hosts. If the next visitor behaves like a random choice from the 28,975 cases in the above table, we can use these counts to define the probabilities. b. Compute the probability that the next visitor makes a purchase and comes from yahoo? What you have computed is the probability p robability of two events happening simultaneously. This is nothing but the joint probability. c. What is the probability that user visits from Google? Have you considered other attributes when computing this value? This is nothing but Marginal probability. A marginal probability is the observing probability of a single attribute, regardless of it’s other attributes. d. Among the visitors from Yahoo, what is the probability of purchase? This probability is nothing but conditional probability. Notice that you restrict re strict only to the Column labelled: Yahoo. So, in the conditional probability, we restrict our sample space as per our requirement. 3. A firm revised its advertising to emphasize the energy-conserving features of its new cars. These ads, as well as ads that do not emphasize conversation were shown to potential customers. The table shows the joint j oint probabilities. Are the reaction and theme independent indepe ndent or dependent? Interpret.
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CSE7315c – Fundamentals of Probability and Statistical Methods
Hint : Compute and check whether Prob (Postive and Conservation) = Prob (Postive) * Prob (Conservation) Customer reaction Theme of advertisement
Negative
Neutral
Positive
Conservation
0.05
0.15
0.3
Standard
0.15
0.15
0.2
4. A manufacturer claims that its drug test will detect steroid use (that is, show positive for an athlete who uses steroids) 95% of the time. What the company does not tell you is that 15% of all steroid-free individuals also test positive ( the false positive rate). 10% of the football team members use steroids. Take E = the event that a football team member tests positive F = the event that a football team member uses steroids Compute the following
P( )
and P( F)
P(E/F) and P( E/F) P(E/ F) and P(!E/!F) 5. In the following table we list some people and the types of laptops and phones they have: Person_id
Laptop
Phone
1
PC
Android
2
PC
Android
3
PC
Android
4
Mac
Iphone
5
Mac
Android
6
Mac
Iphone
7
Mac
Iphone
8
Mac
Android
9
PC
Iphone
Mac
Iphone
10
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CSE7315c – Fundamentals of Probability and Statistical Methods
a. What is the probability that a randomly selected person uses an iPhone? b. What is the probability that a randomly selected person uses an iPhone given that person uses a Mac laptop? c. What is the probability of a person owning a mac given that they own an iphone
6. A retailer uses free prizes to increase the number of shoppers who regularly visits it is stores. Four prize-winning tickets ae hidden in randomly chose items. The total prize of $30,000 is split among the winners. Because some tickets may not be found by end of the contest, the amount won depends on the number of winning tickets that are claimed. Below table defines the random variable W that denotes the number of winners in a future contest. a. Find the Number mean expected value of W of b. Compute the expected value of winning amount winners
P(W=w)
1
0.05
2
0.2
3
0.45
4
0.3
7. Let’s translate what is written below into probability notation. Please match up the English statements in SET – A below with their associated notations in SET – B and write in the probabilities. If there is no English statement matching a pro bability, please write one. SET – A i. We know that only 0.8% of the people in the U.S. have this form of cancer. 99.2% of people don’t have this cancer ii. iii. When the disease is present the test returns a correct POS result 98% of the time; iv. When the disease is present the test returns a incorrect NEG result 2% of v. it returns a correct NEG result 97% of the time in cases When the disease is not present vi. it returns an incorrect POS result 3% of the time in cases when the disease is not present SET – B
a. b. c. d. e. f.
P(POS|cancer) = _______ P(POS|¬cancer) = _______ P(cancer) = _______ P(¬cancer) = _______ P(NEG|cancer) = _______ P(NEG|¬cancer) = _______
8. You need to figure out whether a company is fraud based on the legal charges they filed. We have the knowledge that, the chances a company submitting fraudulent filings is 0.1. There exists an algorithm that can predict fraud. This algorithm returns a correct positive result in 92% of the cases in which the fraud is present, and correct negative results in 90% of the cases
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CSE7315c – Fundamentals of Probability and Statistical Methods
where the fraud is not present. Suppose we observe a company for whom the algorithm test returns a fraud result. Calculate the a posteriori probability that this company truly did fraud in their filings.
9. A certain type of light bulb has an average life of 500 hours, with a standard deviation of 100 hours. The length of life of the bulb can be closely approximated by a normal curve. An amusement park buys and installs 10,000 such bulbs. Find the total number that can be expected to last for each period of time. a. b. c. d. e. f.
At least 500 hours Less than 500 hours Between 350 and 550 hours Between 680 and 780 hours Less than 770 hours More than 440 hours
Hint : Here is the solution for 3(a). You could use the same logic to solve the rest of the exercises. a. At least 500 hours At least 500 hours. Converting into standardized normal variable (Z) Z = (x-µ)/ =500-500/100=0 Prob(x>=500) = Prob(Z>=0)= 1-Prob(Z<0)=1-0.5=0.5 Prob(Z<0) can be obtained from R by using the following R code: pnorm(value of z,mean=0,sd=1)
10. The average lifetime for a car battery of a certain brand is 170 weeks, with a standard deviation of 10 weeks. If the company guarantees the battery for 3 years, what percentage of the batteries sold would be expected to be returned before the end of the warranty period? Assume a normal distribution Hint: Convert 3 years to weeks taking 52 weeks per year. (i.e.; 3*52=156) Compute Prob(X>156) using normal distribution
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