A First Course In Probability, 9th Edition, Indian Version, Sheldon Ross, Chapter 3, Problems, 17.

Question

The probability of 2 children to be twins is given as 2% and the probability of them being identical twins is 0.16%. If both genders are equally likely and identical twins always belong to the same gender, then find the probability that the children are identical twins given that they are both twin boys.

Answer 1

This is the TREE diagram for above problem:

Probability of Twins = $P(Twins)$ = $2$% = $0.02$

Probability of being Identical given that they are twins =

$P(Identical | Twins)$ = $\frac{P(Identical  \cap  Twins)}{P(Twins)}$ = $\frac{0.0016}{0.02}$  = $0.08$ [“Probability of being identical twins = $P(Identical \cap Twins) = 0.16$ % = $0.0016$”]

Now, if they are “$Identical$” then Probability of $Both$ $Boys$(BB) and $Both$ $Girls$(GG) will be same = $0.5$

If they are “$NOT$ $Identical$” then we will have $4$ possibilities as shown in diagram with Probability of $0.25$

$\therefore$ Probability of $Both$ $Boys$ and they are $Twins$ :

$P(Twins \cap BB)$ = $(0.02 \times 0.08 \times 0.5) + (0.02 \times 0.92 \times 0.25)$ = $0.0054$

 $\therefore$ Probability of they are $Identical$ $Twins$ given that they are $Both$ $Twin$ $Boys$ :

$P(Identical \cap Twins | Twins \cap BB )$  = $\frac{0.02 \times 0.08 \times 0.5}{0.0054}$ = $0.148$ = $14.8$ % [Ans]

Answer 2

For 2 children possible outcomes ={ BB,BG,GB,GG} ,P(twins)=0.02,P(identical ∩ twins)=0.0016. Identical twins always belong to the same gender means for this possible outcomes ={BB,GG}.But for non identical twins outcomes { BB,BG,GB,GG} . Now we have to find P{(identical ∩ twins)|(twins ∩ BB)}.

Now we have to find P(twins ∩ BB)=P(twins ∩ identical ∩  BB)+ P(twins ∩ non identical ∩  BB) =P(BB|twins ∩ identical)*P(twins ∩ identical)+P(BB|twins ∩ non identical)*P(twins ∩ non identical)=0.5 * 0.0016+0.25 * 0.92 * 0.02=0.0054

[ P(twins ∩ non identical)=P(non identical | twins)* P(twins)={1-P(identical | twins)}* P(twins)={1-(P(identical ∩ twins)/P(twins))}*P(twins)=

{1-(0.0016/0.02)}*0.02=0.92*0.02       ]

Now P{(identical ∩ twins)|(twins ∩ BB)}=P(twins ∩ identical ∩  BB)/P(twins ∩ BB)=(0.5 * 0.0016)/0.0054=0.1481=14.81%