Thread: Bayes' Theorem Application problem.

Identical twins come from the same egg and hence are the same sex. Fraternal Twins have a 50-50 chance of being the same sex. Among twins, the probability of a fraternal set is p, and an identical set is q=1-p. If the next set of twins are the same sex, what is the probability that they are identical?

I will denote B as the event of the next set of twins are the same sex. I will then denote A as the event of the next set of twins are identical.

So we know that if they are the same sex they are either identical or fraternal.

So then we know that P[A]= q , therefore, P[A']= p= q-1

We want to find P[ A | B ] using Bayes theorem.


But from this I believe I am missing some information here... We need P[ B|A] and P[B|A']

No, that is sufficient information to give an answer in terms of p.

Imagine 1000 sets of twins. 1000p are identical twins and 1000q= 1000- 1000p are fraternal. Of the 1000p identical twins, all 1000p are of the same sex. Of the 1000q= 1000- 1000p fraternal twins, half of them, 500- 500p, are the same sex. So we have a total of 1000p+ 500- 500p= 500p+ 500 pairs off twins that are of the same sex. Of those 500p+ 500 pairs, 1000p were identical twins. Given that a set of twins are of the same sex, the probability they are identical twins is 1000p/(500p+ 500)= 2p/(p+1).

For example, if p= 0.5, that is 1/(3/2)= 2/3. If p= 1/4, that is (1/2)/(5/4)= 2/5.

Originally Posted by math951

Identical twins come from the same egg and hence are the same sex. Fraternal Twins have a 50-50 chance of being the same sex. Among twins, the probability of a fraternal set is p, and an identical set is q=1-p. If the next set of twins are the same sex, what is the probability that they are identical?
I will denote B as the event of the next set of twins are the same sex. I will then denote A as the event of the next set of twins are identical.
So we know that if they are the same sex they are either identical or fraternal.
So then we know that P[A]= q , therefore, P[A']= p= q-1
We want to find P[ A | B ] using Bayes theorem.

$ \begin{align*}\mathcal{P}(A|B)&=\dfrac{\mathcal{P} (A\cap B)}{\mathcal{P}(B)} \\&=\dfrac{ \mathcal{P} (B|A)\mathcal{P}(A)}{\mathcal{P}(B|A)
\mathcal{P}(A)+\mathcal{P}(B|A')\mathcal{P}(A')} \end{align*}$

As far as $\mathcal{P}(B|A')$, we know that given the twins are not identical the probability that they are same sex is what?

Originally Posted by Plato

$ \begin{align*}\mathcal{P}(A|B)&=\dfrac{\mathcal{P} (A\cap B)}{\mathcal{P}(B)} \\&=\dfrac{ \mathcal{P} (B|A)\mathcal{P}(A)}{\mathcal{P}(B|A)
\mathcal{P}(A)+\mathcal{P}(B|A')\mathcal{P}(A')} \end{align*}$

As far as $\mathcal{P}(B|A')$, we know that given the twins are not identical the probability that they are same sex is what?

OHHHH I see.


P[B|A] has to =1 <--- And this is because given the information of A. Because identical twins are always same sex. Now for P[B|A'] we know as you said the twins are not identical, but they are the same sex, so that means the probability is .5 And that is all the information we need to compute.

Just to be clear, when you see P[A|B]

one should read it as, " find the probability of event A, given event B has already happened."

Originally Posted by math951

Just to be clear, when you see P[A|B] one should read it as, " find the probability of event A, given event B has already happened."

Absolutely correct. The idea being that $A$ happens within the event space where $B$ has already happened.
$B$ has conditioned the event $A$.
$\mathcal{P}(A|B)=\dfrac{\mathcal{P}(A\cap B)}{\mathcal{P}(B)}$ states that $A~\&~B$ happen within the space where $B$ has happened.

In this thread, the question was that if same sex twins are born(B), what is the probability that those twins are identical(A)?
Within the space of same sex twins, what is the probability that a randomly selected set of twins are identical:$\mathcal{P}(A|B)$ .