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Question

A fair coin and an unbiased die are tossed. Let A be the event ‘head appears on the coin’ and B be the event ‘3 on the die’. Check whether A and B are independent events or not.

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Solution

It is given that when head appears on the coin, the event is A and when 3 on the die, the event is B.

Total possible outcomes are 12. Sample space is given by,

S={ ( H,1 ),( H,2 ),( H,3 ),( H,4 ),( H,5 ),( H,6 ), ( T,1 ),( T,2 ),( T,3 ),( T,4 ),( T,5 ),( T,6 ) }

The event A can be possible as,

A={ ( H,1 ),( H,2 ),( H,3 ),( H,4 ),( H,5 ),( H,6 ) }

Then, the probability of A is,

P( A )= 6 12 = 1 2

The event B can be possible as,

B={ ( H,3 ),( T,3 ) }

Then, the probability of B is,

P( B )= 2 12 = 1 6

The condition when the head appear on the coin and 3 on the die is ( H,3 ).

The probability is given as,

P( AB )= 1 12 (1)

The formula when two events are independents is given as,

P( AB )=P( A )P( B )

Substitute the value of P( A )= 1 2 and P( B )= 1 6 in the above formula

P( A )×P( B )= 1 2 × 1 6 P( A )×P( B )= 1 12 (2)

Equation (1) and (2) P( AB )=P( A )P( B ).

Hence, A and B are independent events.


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