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A coin is tossed thrice. Let X be the event that head occurs in each of the first two tosses. Let Y be the event that a tail occurs on the third toss. Let Z be the event that two tails occur in three tosses.
Based on the above information, which one of the following statements is TRUE?

1. X and Y are not independent
2. Y and Z are dependent
3. Y and Z are independent
4.  X and Z are independent

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If A and B are independent then P(A/B) = P(A) and P(B/A) = P(B).

ans should be B

event X =HH _

event  Y= _ _T      so X and Y are independent  so option A is out

event Z = 2T 1 H (any order TTH,THT,HTT)  so if X occurs Z can not occur hence X and Z are dependent events so option D is out

event Z = 2T 1 H ( TTH(event Y is not possible hence Y and Z are dependent) ,THT(event Y is possible) ,HTT (event Y is possible)

consider  choice C and D the question is asked about 2 (exactly 2) tails occur in 3 tosses now if there is a tail in 3rd toss then we need only one tail in first 2 tosses else we need 2 tails   hence they are dependent on each other

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Can anyone please elaborate i am finding it difficult to understand.

Z is dependent on X and Y and so option B).

Why Z is dependent on X ???

• If X happens, then Z will never happen (i.e) if first 2 tosses result as Heads,then there can never be 2 tails in 3 tosses.
• If X doesnot happen, then Z happens if first 2 tosses are TAIL-TAIL and it may or maynot happen if first 2 tosses are HEAD-TAIL or TAIL-HEAD.

So the occurence of Z is greatly influenced by whether X has happened or not.

Why Z is dependent on Y ???

• If first 2 tosses are HEAD-TAIL or TAIL-HEAD, then whether Z will happen or not depends on whether Y has happened or not (i.e) if Y happen Z will happen else Z willnot happen.

So Z is an event whose occurence is greatly influenced by the occurences of both X and Y.

Note : X and Y are independent. EX: If 2 coins are tossed, outcome of 1 st toss has nothing to do with 2 nd toss (i.e) outcome of  2 nd toss is independent of 1 st toss. Similiarly here outcome of first 2 tosses has nothing to do with the outcome of the 3 rd toss.