probb. Assignment 1-3_2011 - MA20106(Probability and...

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MA20106 (Probability and Stochastic Processes) Assignments 1-3 2010-11 Assignment-1 1 . Let E , F and G be three events. Find the expressions for the events so that of E , F and G : ( a ) only E occurs ( b ) both E and G but not F occurs ( c ) at least one of the events occurs ( d ) at least two of the events occurs ( e ) all three occurs ( f ) none of the events occurs ( g ) at most one of them occurs ( h ) at most two of them occurs ( i ) exactly two of them occurs ( j ) at most three of them occurs. 2 . Find the simple expression for the following events : ( a ) ( E F )( E F c ) ( b ) ( E F )( E c F )( E F c ) ( c ) ( E F )( F G ) 3 . Show that the probability that exactly one of the events E or F occurs equal P ( E ) + P ( F ) - 2 P ( EF ). 4 . Prove that P ( E c F c ) = 1 - P ( E ) - P ( F ) + P ( EF ). 5 . An urn contains 5 red, 6 blue, and 8 green balls. If a set of 3 balls is randomly selected, what is the probability that each of the balls will be of the same colour? Of different colour? Repeat under the assumption that whenever a ball is selected, its colour is noted and it is then replaced in the urn before the next selection. 6 . Urn A contains 3 red and 3 black balls, whereas urn B contains 4 red and 6 black balls. If a ball is randomly selected from each urn, what is the probability that the balls will be of same colour? 7 . There are 5 hotels in a certain town. If 3 people check into hotels in a day, what is the prob- ability they each check into a different hotels? 8 . If a die is rolled 4 times, what is the probability of 6 coming up at least once? 9 . A group of 6 men and 6 women is randomly divided into two groups of size 6 each. What is the probability that both groups will have the same number of men?
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