4.4 Random Variables, Density Functions

Odd Heads And Matches


We flip 3 coins. Let

\[C:=\#\text{heads}\] \[M:=\begin{cases}1&\text{if all flips match}\\0&\text{otherwise}\end{cases}\] \[O:= \text{ odd #heads}\]

Let \(I_O\) be the indicator variable for \(O\).

We want to show that \(I_O\) and \(M\) are independent. Please enter all answers in the form of decimals with three significant digits.

  1. What is \(\Pr[I_O=1]\)?

    \(\Pr[I_O=1]=\Pr[O]=\Pr[C=1 \text{OR} C=3]=\Pr[C=1]+\Pr[C=3]=\frac{3}{8}+\frac{1}{8}=\frac{1}{2}\)

  2. What is \(\Pr[I_O=0]\)?

    \(\Pr[I_O=0]=1-\Pr[I_O=1]=\frac{1}{2}\)

  3. What is \(\Pr[M=1]\)?

    \(M=1\) if we get all heads or all tails. Hence, \(\Pr[M=1]=\Pr[HHH]+\Pr[TTT]=\frac{1}{8}+\frac{1}{8}=\frac{1}{4}\).

  4. What is \(\Pr[M=1 \text{ AND } I_O=1]\)?

    The only outcome in this event is HHH (all heads).

  5. What is \(\Pr[M=0 \text{ AND } I_O=1]\)?

    The event \([M=0 \text{ AND } I_O=1]\) is equivalent to \(C=1\).

  6. What is \(\Pr[M=1 \text{ AND } I_O=0]\)?

    The only outcome in this event is TTT (all tails).

  7. What is \(\Pr[M=0 \text{ AND } I_O=0]\)?

    The event \([M=0 \text{ AND } I_O=0]\) is equivalent to \(C=2\).

Now you should verify that \(\Pr[M=k_1 \text{ AND } I_O=k_2]=\Pr[M=k_1]\Pr[I_O=k_2]\) for all \(k_1,k_2\in \{0,1\}\).