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Question: A fair coin is tossed n times. If the probability that head occurs 6 times is equal to the probabili...

A fair coin is tossed n times. If the probability that head occurs 6 times is equal to the probability that head occurs 8 times, then n is equal to

A

15

B

14

C

12

D

7

Answer

14

Explanation

Solution

Probability that head occurs 6 times =nC6(12)6(12)n−6= { } ^ { n } C _ { 6 } \left( \frac { 1 } { 2 } \right) ^ { 6 } \left( \frac { 1 } { 2 } \right) ^ { n - 6 } and probability that head occurs 8 times

=nC8(12)8(12)n−8= { } ^ { n } C _ { 8 } \left( \frac { 1 } { 2 } \right) ^ { 8 } \left( \frac { 1 } { 2 } \right) ^ { n - 8 }

∴nC6(12)6(12)n−6=nC8(12)8(12)n−8\therefore { } ^ { n } C _ { 6 } \left( \frac { 1 } { 2 } \right) ^ { 6 } \left( \frac { 1 } { 2 } \right) ^ { n - 6 } = { } ^ { n } C _ { 8 } \left( \frac { 1 } { 2 } \right) ^ { 8 } \left( \frac { 1 } { 2 } \right) ^ { n - 8 }

nC6=nC8{ } ^ { n } C _ { 6 } = { } ^ { n } C _ { 8 } ⇒ (n−6)(n−7)=56⇒n=14( n - 6 ) ( n - 7 ) = 56 \Rightarrow n = 14 .