#1. An urn contains 4 balls, each with a different color: Red, Orange, Yellow, a
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Question
#1. An urn contains 4 balls, each with a different color: Red, Orange, Yellow, and Green. In each of two trials, a single ball is drawn from the urn. X is the event the 1st ball is Red. Y ?s the event the 2nd ball is Red (a) Suppose the balls are drawn with replacement (i.e.,the 1st ball is returned to the urn before the 2nd ball is drawn). Are X and Y independent events? Why? (b) Suppose the balls are drawn without replacement (i.e., the 1st ball is not returned to the urn before the 2nd ball is drawn. Arex and Y independent events? why?Explanation / Answer
The urn contains 4 balls, of different colour each: Red (R) , Orange (O), Yellow (Y) and Green (G)
X= event that first ball is red
Y= event that second balls is red
Independent events: Two events are said to be independent if happeneing of one does not affect the happeneing of the other, (if both are independent in their results)
a) Balls are drawn with replacement;
X= first ball is red, in case X takes place then the red ball is repalced, so in the second turn, all colours have 1 ball each and Y is just like it would be if X took place or not;
X'= first Ball is not red; so that non-red ball of a particular colour is replaced with another ball of the same colour, thus all colours have 1 ball each and at the drawing of second balls, Y will be same as beginning, irrespective of X happened or X' happened
Thus, with replacement, X and Y are independent events, since the first drawn ball is being replaced
b) Without replacement:
X= first ball drawn is red and it is not replaced so there is no red ball present to be drawn in this case. So probability of Y =0
X' = first ball drawn is non-red, but in this case, there will be only 3 balls left from which one is to be drawn in the second turn, so Y will have a probability of = P(Y) = 1/3
Thus, happeneing (or not happeneing) of X affects the happening (or not happeneing) of Y.
Thus, X and Y are NOT INDEPENDENT events
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