problem 10 and problem 11 of bayes theorem for ncert bayes theorem

Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die?

Let E1 be the event that the girl threw 1 or2 or 3,or 4 with the dice .

let E2 be the event that the girl threw 5 or 6 with the die.

let A be the event that the girl obtained exactly one head .


P( E1 ) = ( 4 / 6 ) { the girl threw 1 or2 or 3,or 4 with the dice }

P( E2 ) = ( 2 / 6 ) { the girl threw 5 or 6 with the die }

P( A / E1 ) = ( 2 / 4 ) [ {HT,TH} out of {HT,TH,TT,HH}]

P( A / E2 ) = ( 3 / 8 ) [ {HTT,THT,TTH} out of {HTT,THT,TTH,HHH,TTT,THH,HHT,HTH}]


Required probability = P [ she threw 1, 2, 3 or 4 with the die given that she obtained exactly one head ]

Required probability = P [ E1 / A ]




P ( E1 / A ) = [( 4/6)( 2/4)] / { [( 4/6)( 2/4) ] + [( 2 / 6 )( 3 / 8 )] }


P ( E1 / A ) =  [ (1/3) ] / {[(1/3)]+[1/8]} =  ( 8 / 11 )
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problem 11
A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A?

Let E1 be the event that a selected item was produced by A .

let E2 be the event that a selected item was produced by B.

let E3 be the event that a selected item was produced by C.

let A be the event that the selected item was defective .


P( E1 ) = ( 50 / 100 )

P( E2 ) = ( 30 / 100 )

P( E3 ) = ( 20 / 100 )

P( A / E1 ) = ( 1 / 100 )

P( A / E2 ) = ( 5 / 100 )

P( A / E3 ) = ( 7 / 100 )


Required probability = P [ item was produced by A given that the item is defective ]

Required probability = P [ E1 / A ]


                  
P ( E1 / A ) = [( 50 / 100 )( 1 / 100 )] / { [(50/100)(1/100)]+[(30/100)(5/100)] +[(20/100)(7/100)] }


P ( E1 / A ) =  [ 50 ] / {[50]+[150]+[140]} =  ( 5 / 34 )

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