Question

One hundred identical coins each with probability p of showing up heads are tossed once. If 0 <p< 1 and the probability of heads showing on 50 coins is equal to that of heads showing on 51 coins, then the value of p is

• A. $\frac { 1 } { 2 }$
• B. $\frac { 49 } { 101 }$
• C. $\frac { 50 } { 101 }$
• D. $\frac { 51 } { 101 }$

Answer 1

Answer: (D)

$\frac { 51 } { 101 }$

Answer 2

We have ${ } ^ { 100 } C _ { 50 } p ^ { 50 } ( 1 - p ) ^ { 50 } = { } ^ { 100 } C _ { 51 } p ^ { 51 } ( 1 - p ) ^ { 49 }$ or

$\frac { 1 - p } { p } = \frac { 100 ! } { 51 ! .49 ! } \times \frac { 50 ! .50 ! } { 100 ! } = \frac { 50 } { 51 }$ or $51 - 51 p = 50 p$

Ž $p = \frac { 51 } { 101 }$.