In this example, I am unable to understand how the author arrived at the values present in the last two tables.
What I know: (filling in the claim made by the author)
Claim: If win occurs at W=1, the net loss is -31
Proof: $-1+(-2^{1})+(-2^{2})+(-2^{3})+(-2^{4})=-31$
Claim: If win occurs at W=2, the net loss is -15 Proof: $-1+(-2^{1})+(-2^{2})+(-2^{3})=-15$
Claim: If win occurs at W=3, the net loss is -7 Proof: $-1+(-2^{1})+(-2^{2})=-7$
Claim: If win occurs at W=4, the net loss is -3 Proof: $-1+(-2^{1})=-3$
Summing up gives -63 as is mentioned.
Now, I do not understand part where the author speaks about "the negative values....". His intent is difficult to decipher.
6,5,4,3,2,1 refers to the time L of the loss, I believe
The values that follows below is unclear.
$\space$
the values -1,...,-63 is unclear as to where they were derived from
the values below that follows is also very much unclear as to their origin.
I apologise for the terse description but I've spend hours on this just trying to understanding what the author is trying to convey!
Any help is appreciated.
$$\begin{array}{|c|c|} \hline \text{Combination}&\text{Winnings}\\ \hline WWW&+3\\ \hline WWL&+1\\ \hline WLW&+2\\ \hline WLL&-2\\ \hline LWW&+2\\ \hline LWL&+0\\ \hline LLW&+1\\ \hline LLL&-7\\\hline \end{array}$$
What auther there is saying that winnings of $-63$ occurs $1$ out of $64$ times, $-1$ occurs $3$ times out of $64$ and so on.
As you can see in the example negative winnings are less frequent but are "heavier" in a sense that they cost you more
Moreover, The expected winnings, due to these "heavy" losses is always 0 (assuming P(Win)=P(Loss)=$50\%$
The author arrived at those negative values simply by listing all possible scenarios and counting. (There might be a more sophisticated way but i'm not sure about that)