Question

If a coin is fair, what is the probability that we obtain a) exactly 120 heads in 250 flips b) more than 120 heads in 250 flips c)less than 120 heads in 250 flips?

Answer 1

Solution:

n = 250

p = 1/2 = 0.5

Formula:

P_{(k out of n )}= n!*p^k * q^n-k / k! *(n - k)!

( a )

P( x = 120 ) = 250!*0.5¹²⁰ * 0.5^250-120 / 120! *(250 - 120 )!

= 0.0413

( b )

P( x > 120 ) = 250!*0.5¹²¹ * 0.5^250-121 / 121! *(250 - 121 )! + 250!*0.5¹²² * 0.5^250-122 / 122! *(250 - 122 )!+

........+ 250!*0.5²⁵⁰ * 0.5^250-250 / 250! *(250 - 250 )!

= 0.7153

( c )

P( x < 120 ) = 250!*0.5¹¹⁹ * 0.5^250-119 / 119! *(250 - 119 )! + 250!*0.5¹¹⁸ * 0.5^250-118 / 118! *(250 - 118 )!+

........+ 250!*0.5⁰ * 0.5^250-0 / 0! *(250 - 0 )!

= 0.2433