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A man is known to speak truth 3 out of 5 times. He throws a die and reports that it is a one. Find the probability that it is actually a one

Asked by Deepak Singh | 17th Nov, 2015, 11:11: PM

Expert Answer:

$L e t space E space b e space t h e space e v e n t space t h a t space t h e space m a n space r e p o r t s space t h a t space o n e space o c c u r s space i n space t h e space t h r o w i n g space o f space t h e space d i e space a n d L e t space A space b e space t h e space e v e n t space t h a t space 1 space o c c u r s space a n d space B space b e space t h e space e v e n t space t h a t space 1 space d o e s space n o t space o c c u r. P open parentheses A close parentheses equals P r o b a b i l i t y space t h a t space o n e space o c c u r s equals 1 over 6 P open parentheses B close parentheses equals P r o b a b i l i t y space t h a t space o n e space o c c u r s equals 5 over 6 P open parentheses E vertical line A close parentheses equals P r o b a b i l i t y space t h a t space t h e space m a n space r e p o r t s space t h a t space o n e space o c c u r s space w h e n space o n e space h a s space a c t u a l l y space o c c u r r e d space o n space t h e space d i e space space space space space space space space space space space space space equals P r o b a b i l i t y space t h a t space t h e space m a n space s p e a k s space t h e space t r u t h equals 3 over 5 P open parentheses E vertical line B close parentheses equals P r o b a b i l i t y space t h a t space t h e space m a n space r e p o r t s space t h a t space o n e space o c c u r s space w h e n space o n e space h a s space n o t space a c t u a l l y space o c c u r r e d space o n space t h e space d i e space space space space space space space space space space space space space equals P r o b a b i l i t y space t h a t space t h e space m a n space d o e s space n o t space s p e a k space t h e space t r u t h equals 1 minus 3 over 5 equals 2 over 5 T h u s comma space b y space B a y e apostrophe s space T h e o r e m comma space w e space h a v e comma P open parentheses A vertical line E close parentheses equals P r o b a b i l i t y space t h a t space t h e space r e p o r t space o f space t h e space m a n space t h a t space o n e space h a s space o c c u r r e d space i s space a c t u a l l y space a space o n e space space space space space space space space space space space space equals fraction numerator P open parentheses A close parentheses cross times P open parentheses E vertical line A close parentheses over denominator P open parentheses A close parentheses cross times P open parentheses E vertical line A close parentheses plus P open parentheses B close parentheses cross times P open parentheses E vertical line B close parentheses end fraction space space space space space space space space space space space space space space space equals fraction numerator 1 over 6 cross times 3 over 5 over denominator 1 over 6 cross times 3 over 5 plus 5 over 6 cross times 2 over 5 end fraction space space space space space space space space space space space space space space space equals fraction numerator 3 over 30 over denominator 3 over 30 plus 10 over 30 end fraction space space space space space space space space space space space space equals fraction numerator 3 over 30 over denominator 13 over 30 end fraction space space space space space space space space space space space space equals 3 over 13$

Answered by Vimala Ramamurthy | 18th Nov, 2015, 10:21: AM

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