Exercise 1: A fair six-sided die is tossed until a 6 is observed. Let X be the cardinal of tosses until (and including) the aboriginal 6 is observed.
a) Acquisition the anticipation that at atomic 5 rolls are appropriate to access the aboriginal 6
b) Accustomed that a 6 was not formed on the aboriginal 3 rolls, acquisition the anticipation that at atomic 5 added rolls are bare to access the aboriginal 6.
c) Your two answers should be equal. What acreage of the geometric accidental capricious is illustrated in this exercise?
Exercise 2: Consider a apprentice demography a multiple-choice test. On a accustomed question, either the apprentice knows the answer, in which case he answers it correctly, or he does not apperceive the answer, in which case he guesses acquisitive to assumption the actual answer. Accept there are bristles multiple-choice alternatives. Let p be the anticipation that the student
knows the actual answer. Let us accept that the anticipation that the apprentice gets the actual acknowledgment accustomed that he guesses is 1/5. Show that the anticipation that the student
knowsthe actual acknowledgment accustomed that the apprentice gotthe actual acknowledgment is 5p/ 4p+1
HINTUse the Theorem of Total Anticipation calm with Bayes' Theorem. Carefully ascertain the events.
Exercise 3: In this game, there are bristles fair six-sided die. Begin by rolling all bristles dice (round 1). The cold is to access bristles 6s in absolutely two rounds. For example, if two 6s were formed in the aboriginal round, afresh you cycle the actual three dice in the additional annular in the hopes of rolling three 6s. If alone one 6 was formed in the aboriginal round, you cycle the actual four dice in the additional annular in the hopes of rolling four 6s. If no 6s are formed in the aboriginal round, afresh all bristles dice are formed in the additional annular in the hopes of rolling bristles 6s. And if bristles 6s are formed in the aboriginal round, the bold is over! But you don't win because it did not booty you absolutely two circuit to access the bristles 6s (and you are not accustomed to try again). Use the Theorem of Total Anticipation to acquisition the anticipation of rolling bristles 6s in absolutely two rounds. You will accomplish acceptable use of the binomial anticipation distribution.
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