Question
Tue March 17, 2015 By: Yashasvi Vyas

How many times must a man toss a fair coin , so that the probability of having atleast one head is more than 80% ?

Expert Reply
Vimala Ramamurthy
Tue March 17, 2015
T h e space p r o b a b i l i t y space o f space h a v i n g space h e a d space w h i l e space t o s sin g space a space c o i n space i s space 1 half S i n c e space t h e r e space a r e space o n l y space t w o space o u t space c o m e s space apostrophe h e a d apostrophe space o r space apostrophe t a i l apostrophe space i t space f o l l o w s space t h e space apostrophe s u c c e s s apostrophe space o r space apostrophe f a i l u r e apostrophe space p r o p e r t i e s o f space b i n o m i a l space d i s t r i b u t i o n. L e t space X space b e space d e n o t e space t h e space n u m b e r space o f space s u c c e s s e s space i n space n space t r i a l s. T h u s space t h e space p r o b a b i l i t y space o f space g e t t i n g space r space h e a d s space w i t h space n space c o i n s space i s space g i v e n space b y P open parentheses X equals r close parentheses equals C presuperscript n subscript r cross times open parentheses 1 half close parentheses to the power of n P r o b a b i l i t y space o f space g e t t i n g space a t l e a s t space o n e space h e a d equals P open parentheses X equals 1 close parentheses plus P open parentheses X equals 2 close parentheses plus P open parentheses X equals 3 close parentheses plus... plus P open parentheses X equals n minus 1 close parentheses G i v e n space t h a t P open parentheses X equals 1 close parentheses plus P open parentheses X equals 2 close parentheses plus P open parentheses X equals 3 close parentheses plus... plus P open parentheses X equals n minus 1 close parentheses greater than 80 over 100 A d d space P open parentheses X equals 0 close parentheses space o n space b o t h space t h e space s i d e s comma space w e space h a v e comma P open parentheses X equals 0 close parentheses plus P open parentheses X equals 1 close parentheses plus P open parentheses X equals 2 close parentheses plus P open parentheses X equals 3 close parentheses plus... plus P open parentheses X equals n minus 1 close parentheses greater than P open parentheses X equals 0 close parentheses plus 80 over 100 rightwards double arrow 1 greater than P open parentheses X equals 0 close parentheses plus 4 over 5 rightwards double arrow 1 minus 4 over 5 greater than P open parentheses X equals 0 close parentheses rightwards double arrow 1 fifth greater than P open parentheses X equals 0 close parentheses rightwards double arrow P open parentheses X equals 0 close parentheses less than 1 fifth rightwards double arrow C presuperscript n subscript 0 cross times open parentheses 1 half close parentheses to the power of n less than 1 fifth rightwards double arrow open parentheses 1 half close parentheses to the power of n less than 1 fifth T h e space a b o v e space i n e q u a l i t y space h o l d s space o n l y space i f space n equals 3 T h u s space t h e space m a n space t o s s e s space t h e space c o i n space t h r e e space t i m e s.
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