Question

Hat #1 contains four gold coins, one silver coin and five copper coins. Hat #2 contains three gold coins, six silver coins and one copper coin. (A) You will randomly select one coin from each of the two hats. The outcome of interest is the colour of each of the selected coins. Give the complete sample space of possible outcomes, and calculate the probability of each outcome. (B) What is the probability that the two selected coins are the same colour? (C) Let X be the number of gold coins that are selected. Find the probability distribution of X.

Answer 1

#2 )

Hat 1 = { G₁ = 4 , S₁ = 1 , C₁ = 5 } , Hat 2 = { G₂ = 3 , S₂ = 6 , C₂ = 1}

A) Sample space = { G₁ G₂ } , { G₁ S₂ }, { G₁ C₂ }, { S₁ G₂ }, { S₁ S₂ }, { S₁ C₂ }, { C₁ G₂ }, {C₁ S₂ }, { C₁ C₂ }

P( G₁ G₂ ) = (4/10)*(3/10) = 0.4*0.3 = 0.12

P( G₁ S₂ ) = ( 4/10)*( 6/10 ) = 0.4*0.6 = 0.24

P( G₁ C₂ ) = ( 4/10)*( 1/10 ) = 0.4*0.1 = 0.04

P( S₁ G₂ ) = ( 1/10)*( 3/10 ) = 0.1*0.3 = 0.03

P( S₁ S₂ ) = ( 1/10)*( 6/10 ) = 0.1*0.6 = 0.06

P( S₁ C₂ ) = ( 1/10)*( 1/10 ) = 0.1*0.1 = 0.01

P( C₁ G₂ ) = ( 5/10)*( 3/10 ) = 0.5*0.3 = 0.15

P( C₁ S₂ ) = ( 5/10)*( 6/10 ) = 0.5*0.6 = 0.30

P( C₁ C₂ ) = ( 5/10)*( 1/10 ) = 0.5*0.1 = 0.05

B) P( two selected coins are the same colour ) = P( G₁ G₂ ) + P( S₁ S₂ ) + P( C₁ C₂ ) = 0.12 + 0.06 + 0.05 = 0.23

C) X = number of gold coins that are selected

X = { 0,1,2 )

For X = 0 ; P ( X = 0 ) = P( C₁ C₂ ) + P( C₁ S₂ ) +P( S₁ S₂ ) +P( S₁ C₂ ) = 0.05 + 0.30 + 0.06 + 0.01 = 0.42

For X = 1 ; P ( X = 1 ) = P( G₁ C₂ ) + P( G₁ S₂ ) +P( S₁ G₂ ) +P( C₁ G₂ ) = 0.24 + 0.04 + 0.03 + 0.15 = 0.46

For X = 2 ; P ( X = 2 ) =  P( G₁ G₂ ) = 0.12