Question

A biased die is twice as likely to show an even number as an odd number. If such a die is thrown twice, find the probability distribution of the number of sixes. Also, find the mean of the distribution.

Answer 1

Let the probability for odd Number = p

and probability for even number = 2p

⇒ P + 2P = 1   ...`[∴ "P"("E") + "P"(bar"E") = 1)`

⇒ 3P = 1

`P = 1/3`

Probability for odd No = `1/3`

Probability for even No = `2/3`

Probability Distribution is X

`P(X=0) =  ^2C_0 (1/3)^(2-0)(2/3)^0 = 1xx1/9 = 1/9`

`P(X=1) =  ^2C_1 (1/3)^(2-1) (2/3)^1`

`= 2xx1/3xx2/3 = 4/9`

`P(X=2) =  ^2C_2(1/3)^(2-2) (2/3)^2`

`= 1xx1xx4/9 = 4/9`

X	0	1	2
P(X)	`1/9`	`4/9`	`4/9`
XP(x)	0	`4/9`	`8/9`

Mean = `sump_ix_i = 12/9`