Statistics- A new father needs to shop for baby bottles

| March 30, 2017

Question
1.We toss a coin three times and choose S = {H, T}3 for our sample space. Construct an exampleof three events that are pairwise independent but not independent. Write out your calculations justifying that they are pairwise independent but not independent.

2.A new father needs to shop for baby bottles.There are three major baby bottle brands.Theonline product rating for a brand A, B, and C bottle is 4, 2, and 3 stars, respectively.Heinterprets each star as a probability of 1/5 that his baby likes the bottle brand, meaning thathis baby likes the A, B, and C bottles with probability 4/5, 2/5, and 3/5, respectively.

He is unsure which bottle brand is being sold at the local store, but he knows that will theyonly carry one bottle brand due to contractual obligations in the highly competitive baby bottlemarket.Based on a quick national market share analysis, he finds out that the probability offinding a bottle of brand A, B, and C at this store is 2/10, 3/10, and 5/10 respectively.

What is the probability that the baby likes the brand from the local store?

3.It is the year 2017, and you can buy cheap tests for the Zika virus at your local pharmacy.Although the test is very reliable, it can lead to a misdiagnosis. If you test positive, then youneed to see a doctor to confirm your diagnosis via bloodwork. The test diagnoses the Zika viruscorrectly for 98% of the people with the virus. There is a 2% chance you test positive if youdon’t have the virus.
Back in 2016, when the test was developed, it was part of a clinical trial with a group of peoplefor whom it was known that 50% had the virus.
(a) What is the probability that an individual who tested positive in the clinical trial had

contracted the Zika virus?
(b) Back in November 2016, the politician Bernary Truzio ran for President when the test cameon the market. He was so encouraged by his answer to part (a) that he proposed to runthe test on the whole population. Only 0.1% of the population had the virus. What is theprobability that an individual who tested positive had contracted the Zika virus?
(c) Give an intuitive explaination to Bernary Truzio why his proposal doesn’t have the desiredeffect.
(d) He buys your argument. Since he knows that 10% of traffic accidents are caused by driversunder the influence and 90% by drivers sober, he next proposes to only allow drunkendrivers on the roads. What would you say to him?

4.IEOR offers a new class entitled ”Probability for enthusiasts”. We randomly select a studentfrom this class. The probability that the student is from the Northeast is 4/5, and the probability

that (s)he is from Europe is 1/10. If the student is from the Northeast, the student is female withprobability 1/4. If the student is not from the Northeast, the student is male with probability1/2. All European students in the class are males. Compute the probability that:
(a) the student is male and from the Northeast.(b) the student is from the Northeast given that the student is male.

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