Question

In a particular town 10% of the families have no children, 20% have one child, 40% have two children, 20% have three children, and 10% have four. Let T represent the total number of children, and G the number of girls, in a family chosen at random from this town. Assuming that children are equally likely to be boys or girls, find the distribution of G. Display your answer in a table and sketch the histogram.

Answer 1

G, be number of girl child in family

T be total number of children

P[ T = 0 ] = 10% = 0.1

P[ T = 1 ] = 10% = 0.2

P[ T = 2 ] = 10% = 0.4

P[ T = 3 ] = 10% = 0.2

P[ T = 4 ] = 10% = 0.1

G can take values 0,1,2,3,4

P[G = 0] = P[ G = 0 , T = 0 ] + P[ G = 0 , T = 1 ] + P[ G = 0 , T = 2 ] + P[ G = 0 , T = 3 ] + P[ G = 0 , T = 4 ]

P[G = 0] = P[ G = 0 | T = 0 ]*P[ T = 0 ] + P[ G = 0 | T = 1 ]*P[ T = 1 ] + P[ G = 0 | T = 2 ]*P[ T = 2 ] + P[ G = 0 | T = 3 ]*P[ T = 3 ] + P[ G = 0 | T = 4 ]*P[ T = 4 ]  

P[ G = 0 | T = 0 ] = family having no children and no girl child = 1

P[ G = 0 | T = 1 ] = family having one child and no girl child = 0.5 ( equal probability for kid being girl or boy )

P[ G = 0 | T = 2 ] = family having two children and no girl child = 0.5*0.5 (both boys, equal probability for kid being girl or boy )

P[ G = 0 | T = 3 ] = family having three children and no girl child = 0.5*0.5*0.5

P[ G = 0 | T = 4 ] = family having four children and no girl child = 0.5*0.5*0.5*0.5

P[G = 0] = 1*0.1 + 0.5*0.2 + 0.5*0.5*0.4 + 0.5*0.5*0.5*0.2 + 0.5*0.5*0.5*0.5*0.1 = 0.1 + 0.1 + 0.1 + 0.025 + 0.00625 = 0.33125

P[G = 1] = P[ G = 1 , T = 0 ] + P[ G = 1 , T = 1 ] + P[ G = 1 , T = 2 ] + P[ G = 1 , T = 3 ] + P[ G = 1 , T = 4 ]

P[G = 1] = P[ G = 1 | T = 0 ]*P[ T = 0 ] + P[ G = 1 | T = 1 ]*P[ T = 1 ] + P[ G = 1 | T = 2 ]*P[ T = 2 ] + P[ G = 1 | T = 3 ]*P[ T = 3 ] + P[ G = 1 | T = 4 ]*P[ T = 4 ]  

P[ G = 1 | T = 0 ] = family having no children and one girl child = 0

P[ G = 1 | T = 1 ] = family having one child and one girl child = 0.5 ( equal probability for kid being girl or boy )

P[ G = 1 | T = 2 ] = family having two children and one girl child = 0.5*0.5*2 (one girl and one boy, equal probability for kid being girl or boy in two ways  )

P[ G = 1 | T = 3 ] = family having three children and one girl child = 0.5*0.5*0.5*3 ( 3 ways )

P[ G = 1 | T = 4 ] = family having four children and one girl child = 0.5*0.5*0.5*0.5*4 ( 4 ways )

P[G = 1] = 0*0.1 + 0.5*0.2 + 0.5*0.5*2*0.4 + 0.5*0.5*0.5*0.2*3 + 0.5*0.5*0.5*0.5*0.1*4 = 0.1 + 0.2 + 0.075 + 0.025 = 0.4

P[G = 2] = P[ G = 2 , T = 0 ] + P[ G = 2 , T = 1 ] + P[ G = 2 , T = 2 ] + P[ G = 2 , T = 3 ] + P[ G = 2, T = 4 ]

P[G = 2] = P[ G = 2 | T = 0 ]*P[ T = 0 ] + P[ G = 2 | T = 1 ]*P[ T = 1 ] + P[ G = 2 | T = 2 ]*P[ T = 2 ] + P[ G = 2 | T = 3 ]*P[ T = 3 ] + P[ G = 2 | T = 4 ]*P[ T = 4 ]  

P[ G = 2 | T = 0 ] = family having no children and two girl child = 0

P[ G = 2 | T = 1 ] = family having one child and two girl child = 0

P[ G = 2 | T = 2 ] = family having two children and two girl child = 0.5*0.5

P[ G = 2 | T = 3 ] = family having three children and two girl child = 0.5*0.5*0.5*3

P[ G = 2 | T = 4 ] = family having four children and two girl child = 0.5*0.5*0.5*0.5*6 ( in 4C2 ways )

P[G = 2] = 0*0.1 + 0*0.2 + 0.5*0.5*0.4 + 0.5*0.5*0.5*0.2*3 + 0.5*0.5*0.5*0.5*0.1*6 = 0.1 + 0.075 + 0.0375 = 0.2125

P[G = 3] = P[ G = 3 , T = 0 ] + P[ G = 3 , T = 1 ] + P[ G = 3 , T = 2 ] + P[ G = 3 , T = 3 ] + P[ G = 3, T = 4 ]

P[G = 3] = P[ G = 3 | T = 0 ]*P[ T = 0 ] + P[ G = 3 | T = 1 ]*P[ T = 1 ] + P[ G = 3 | T = 2 ]*P[ T = 2 ] + P[ G = 3 | T = 3 ]*P[ T = 3 ] + P[ G = 3 | T = 4 ]*P[ T = 4 ]  

P[ G = 3 | T = 0 ] = family having no children and three girl child = 0

P[ G = 3 | T = 1 ] = family having one child and three girl child = 0

P[ G = 3 | T = 2 ] = family having two children and three girl child = 0

P[ G = 3 | T = 3 ] = family having three children and three girl child = 0.5*0.5*0.5

P[ G = 3 | T = 4 ] = family having four children and three girl child = 0.5*0.5*0.5*0.5*4

P[G = 3] = 0*0.1 + 0*0.2 + 0*0.4 + 0.5*0.5*0.5*0.2 + 0.5*0.5*0.5*0.5*0.1*4 = 0.025 + 0.025 = 0.05

P[G = 4] = P[ G = 4 , T = 0 ] + P[ G = 4 , T = 1 ] + P[ G = 4, T = 2 ] + P[ G = 4 , T = 3 ] + P[ G = 4, T = 4 ]

P[G = 4] = P[ G = 4 | T = 0 ]*P[ T = 0 ] + P[ G = 4 | T = 1 ]*P[ T = 1 ] + P[ G = 4 | T = 2 ]*P[ T = 2 ] + P[ G = 4 | T = 3 ]*P[ T = 3 ] + P[ G = 4 | T = 4 ]*P[ T = 4 ]  

P[ G = 4 | T = 0 ] = family having no children and four girl child = 0

P[ G = 4 | T = 1 ] = family having one child and four girl child = 0

P[ G = 4 | T = 2 ] = family having two children and four girl child = 0

P[ G = 4 | T = 3 ] = family having three children and four girl child = 0

P[ G = 4 | T = 4 ] = family having four children and four girl child = 0.5*0.5*0.5*0.5

P[G = 4] = 0*0.1 + 0*0.2 + 0*0.4 + 0*0.2 + 0.5*0.5*0.5*0.5*0.1 = 0.00625

Distribution of G

P[ G = 0 ] = 0.33125

P[ G = 1 ] = 0.4

P[ G = 2 ] = 0.2125

P[ G = 3 ] = 0.05

P[ G = 4 ] = 0.00625