Question

Find the probability of throwing at most 2 sixes in 6 throws of a single die.

Answer 1

It is given that a die is thrown 6 times.

Let the number of times of getting six in 6 throws of the die be represented by X.

The probability of getting six in a single throw of die is,

p= 1 6

So,

q=1−p =1− 1 6 = 5 6

Then, X has a binomial distribution with n=6 and p= 1 6 .

The probability of x successes P( X=x ) is,

P( X=x )= C n x q n−x p x = C 6 x ( 5 6 ) 6−x ( 1 6 ) x

Where x=0,1,…,n

The probability of getting at most 2 sixes is P( X≤2 ),

P( X=0 )+P( X=1 )+P( X=2 )= C 6 0 ( 5 6 ) 6 + C 6 1 ( 5 6 ) 5 ( 1 6 )+ C 6 2 ( 5 6 ) 4 ( 1 6 ) 2 =1 ( 5 6 ) 6 +6( 1 6 ) ( 5 6 ) 5 +15( 1 36 ) ( 5 6 ) 4 = ( 5 6 ) 6 + ( 5 6 ) 5 + ( 5 6 ) 4 ( 5 12 ) = ( 5 6 ) 4 [ ( 5 6 ) 2 +( 5 6 )+ 5 12 ]

Simplify further.

P( X≤2 )= ( 5 6 ) 4 [ 25 36 + 5 6 + 5 12 ] = ( 5 6 ) 4 [ 25+30+15 36 ] = ( 5 6 ) 4 ( 70 36 ) = ( 5 6 ) 4 ( 35 18 )

Therefore, the probability of getting at most 2 sixes is ( 5 6 ) 4 ( 35 18 ).