0

I'm having trouble understanding the answer to the following question.

enter image description here

The part I'm particularly stuck on is the jump from step one and two where these two formulas appear to be equated

$Pr[(B_1 \cap B_2) \cap (B_1 \cup B_2)] = Pr[B_1 \cap B_2]$

I can't think of any properties of $\cap$ and $\cup$ that can allow us to do this. I tried drawing a Venn-diagram but was unable to represent $\cup$. Any clarification would be much appreciated!

EDIT:

I saw another solution to this question that simply does this.

Sample space:

BB BG GB GG

Since we already know one child is a boy, we can restrict our sample space to the first 3 entries. Out of those first 3 entries, only 1 entry is BB. The reason I was confused about this solution was that I didn't know the order of the boy and girl mattered. I would have made the "mistake" of making the sample space BB, GG, BG because BG is the same as GB. Please tell me why I'm wrong

Carpetfizz
  • 1,133
  • 1
    Note that $A\cap B \subseteq A\subseteq A\cup B$ and that whenever $E\subseteq F$ then $E\cap F=E$. Apply that where instead of $E$ you use $A\cap B$ and instead of $F$ you have $A\cup B$ and arrive at your result. – JMoravitz May 15 '17 at 04:25
  • 1
    As for the content of the question about the boy-girl paradox... this has been answered several times on this site already, for example this one. As for why order of boy and girl mattering, the two children are distinct and through some arbitrary method you may choose one of the children to be the "first" child and the other to be the "second" (often by age, but if twins then by some other method). – JMoravitz May 15 '17 at 04:30
  • 1
    If you instead insist on calling $BG$ the same as $GB$, then recognize that your sample space is not equiprobable since $BG$ occurs twice as often as $BB$ as well as twice as often as $GG$, so this discrepancy should be accounted for in your calculations. – JMoravitz May 15 '17 at 04:31
  • @JMoravitz thank you for the information and the links to similar questions. I will read them and update / close my post if necessary – Carpetfizz May 15 '17 at 04:35
  • I don't understand how "A and B is a subset of A is a subset of A and B" or if I'm reading that incorrectly – Carpetfizz May 15 '17 at 05:18
  • Show that $A\cap B\subseteq A$ and that $A\subseteq A\cup B$ – JMoravitz May 15 '17 at 15:38

1 Answers1

1

$(B_1 \cap B_2)$ represent the event that both children are boys.

$(B_1 \cup B_2)$ represent the event that at least one of them is a boy. (if both will be girls then this shows that events $B_1$ and $B_2$ will not occur )

And hence,

$(B_1 \cap B_2) \cap (B_1 \cup B_2)$ shows the event that both children are boys.

P.S. : { $BB$ } $\cap$ { $BG$ , $GB$ , $BB$ } $=$ { $BB$ }

Sahil Kumar
  • 751