Question

An urn contains 100 balls that have the numbers 1 to 100 painted on them (every ball has a distinct number). You keep sampling balls uniformly at random (i.e., every ball is equally likely to be picked), one at a time, and without replacement. For 1 ≤ i < j ≤ 100, let Ei,j denote the event that the ball with number j was picked after the ball with number i got picked. Identify which of the following sets of events are independent and which are not :

(a) E7,13 and E41,79

(b) E7,13 and E13,8

(c) (E7,13 ∩ E13,8) and E8,7

Please show work

Answer 1

Total number of balls = 100

given,

let Ei,j denote the event that the ball with number j was picked after the ball with a number I got picked.

Look at option (a)

Since there are showing 2 balls are chosen at an event E 7,13 and event E41, 79

Therefore the total number of ways of choosing the two balls at E7,13 is:

the number of ways of being chosen at E41, 79 is: (because two spots are already covered in choosing E7,13)

now, we have to remain 96 balls which can be arranged in 96 spots:  

in general,

for any i and j ball

Since we have to check whether it is independent or not:

Therefore, Bothe the events are independent

since

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(b) Now, E7,13 means first ball is 7 and next one is 13

E13,8 means the first ball is 13 and next one is 8

Here, we have to choose a total of 3 balls(7,13,8 ) one by one and 97 balls can be arranges in remaining 97 spots.

so, total ways of doing that:

Thus, for checking independency we'll calculate probability:

since we know that we have already calculated that each event probability for any i,j the probability is  

Therefore,

Therefore, both events aren't independent.

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(C)

No. of ways of selecting balls 7, 13, 8 is =  

remaining are 97 balls and 97 spots

now, ways for selecting two balls 8, 7 =

so, total probability =

For independency,

   , already proved above

already proved above for any i, j balls and a single event

Therefore,

Therefore, both events are not independent.

NOTE: for the independency of any two events (A, B) :

P(A, B) = P(A) * P(B)

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