Question

20.21.24.25

are dice 16. Suppose you roll two fair dice. Give an example of two independent events. Given an example of two disjoint events ' (AMB) 74 73. 17. Under what circumstances could two events be both disjoint and independent? 18. Suppose that there are 10 sodas in a cooler: 7 regular and 3 diet. What is the probability that 2 diet sodas are selected at random? What is the probability that 2 diet sodas are selected given at least one diet soda was selected? (10) [ed/at but andet? 19. Suppose your friend flips a fair coin 8 times. She tells you that at least 7 of the flips were heads. What is the probability they were all heads? 20. How would you define expected value to a friend who doesn't know what probability is? 21. How would you define variance to a friend who hasn't taken probability? 22. 5 quarters and 2 nickels are in a bag and two are selected at random. What is the expected amount of money that you will draw out? 23. A multiple choice exam consists of 72 questions. If there are 4 answer choices, only one of which is correct. how many questions would you expect to get right by guessing? 24. Suppose this exam subtracts 1/3 of a point for each wrong answer. What is the expected number of points you would get by guessing (each correct answer is worth 1 point)? What strategy would you suggest for taking this exam? 25. Suppose there is an exam with 50 true/false questions, and 50 multiple choice questions, each with five possible answers (and only one correct answer). What is the expected number of questions you would get right from guessing?

Answer 1

20 . Expected value : In simple it is long term average of random variable X.

For example : If you are playing a die game

rules are like this : if die shows 4 you will get 150 bucks

if not 4 you will loss 10 bucks

So how much do you make in a long run whether you will make profit or gain this can be answered by expected value.if expected value is -ve then it is benefited to casino and if it is positive it is benefited to you(player).

Here what is the chance of getting 4 is 1/6

therefore not getting 4 is 5/6

Then EV = 1/6*150+ 5/6*-10 = 16.67 Since it is positive value in long run you will be benefited.

21 . Variance :

Consider the height of a human being, suppose it has a mean of 1.7 meters. But nobody you actually meet has an exact height of 1.7 meter. Every deviates a bit from this mean even though it is expected height of any one you see. What the variance (or better the still the standard deviation) measures is the probability that random person you meet will be this much away from the mean.

If the variance is tiny, then there is almost no way you will meet someone who is 1.9 and above or 1.5 and below. On the other hand, if the variance is huge, there high chance to meet someone who is 2.0 or 1.4.

24. P(Correct) = 1/2 ,P(Wrong) =1/2

Assumption : took equal probability for correct and wrong.

if question is correct you will get 1 mark if it is wrong you will lose 1/3 marks.

This exam has 25 questions.

Expected no of marks for guessing all 25 questions is : 25*(1/2*1-1/2*1/3) = 8.34

25 . Suppose there is an exam with 50 True/False and 50 multiple choice questions with 5 options.

P(C) = 1/2 and P(W) = 1/2 for true/false questions

P(c) = 1/5 and P(w) = 4/5 for MCQ's

Expected no of questions getting correct = 50*P(C) + 50 *P(c) =50*1/2 + 50*1/5 = 25 + 10 = 35 questions