Consider an experiment that involves rolling a fair six-sided twice. Let X denot
ID: 3231225 • Letter: C
Question
Consider an experiment that involves rolling a fair six-sided twice. Let X denote the number rolled on the first roll and let Y denote the number on the second. (a) How many possible outcomes are there? That is, how many different observed values for (X, Y) can eventuate from this experiment? (Just an answer is okay if you know it). (b) For the remainder of this question you may assume that these outcomes are all equally likely to occur. What is the probability associated with each outcome occurring? (c) Let A denote the event that the sum of the two numbers rolled is less than or equal to 4 (i.e. X + Y lessthanorequalto 4). Calculate the probability of event A occurring. (d) What is the probability of A not occurring? (e) Now, let B denote the event that at least one of the numbers rolled is a 2. Calculate the probability of event B occurring. (f) Are events A and B mutually exclusive? Explain. (g) Calculate the following: i. The probability that both event A and event B occur at the same time. ii. The probability that either event A or event B occurs (this includes the possibility of both occurring). (h) Using one of the probabilities in part (g) above, calculate the probability of neither event A or B occurs. That is, calculate the event that both A and B occur at the same time.Explanation / Answer
d) Lets calcualte tthe probability of A occuring first
Total number of pairs whose sum is less than 4 are :
Total n0 of pairs are 36
Total 6 pairs whic hahve sum less than or equal to 4, therefore P(A) = 6 /36 = 1/6
Probability of A not occuring = 1-1/6 = 5/6
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e)Lets take the cases where 2 appears
Total 11 cases are there ,
so probability will be =11/36
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f)Mutually exclusive" means of any two possible outcomes A and B, the logical expression A and B cannot occur. Put another way, two propositions are mutually exclusive if the following are true: "If A, then not B" and "If B, then not A.
So in this case they arent mutually exhaustive since , few pairs are common
(1,1) (2,1) (3,1) (1,2) (2,2) (1,3)