Fogelin's Palace in Border, Nevada, offers the following unusual bet. If you win, you make a 50 percent profit on your bet; if you lose, you take a 40 percent loss. That is, if you bet $$\$ 1$$ and win, then you get back $$\$ 1.50$$; if you bet $$\$ 1$$ and lose, you get back $$\$ 0.60$$. The chances of winning are fifty-fifty. This sounds like a marvelous opportunity, but there is one hitch: To play, you must let your bet ride with its winnings, or losses, for four plays. For example, with $$\$ 100$$, a four-bet sequence might look like this:
$$
\begin{array}{ccccc}
& \text { Win } & \text { Win } & \text { Lose } & \text { Win } \\
\hline \text { Total } & \$ 150 & \$ 225 & \$ 135 & \$ 202.50
\end{array}
$$
At the end of this sequence, you can pick up $$\$ 202.50$$, and thus make a $$\$ 102.50$$ profit. It seems that Fogelin's Palace is a good place to gamble, but consider the following argument on the other side. Because the chances of winning are fifty-fifty, you will, on the average, win half the time. But notice what happens in such a case:
$$
\begin{array}{lllll}
& \text { Win } & \text { Lose } & \text { Lose } & \text { Win } \\
\hline \text { Total } & \$ 150 & \$ 90 & \$ 54 & \$ 81
\end{array}
$$
So, even though you have won half the time, you have come out $\$ 19$ behind.
Surprisingly, it does not matter what order the wins and losses come in; if two are wins and two are losses, you come out behind. (You can check this.) So, because you are only going to win roughly half the time, and when you win half the time you actually lose money, it now seems to be a bad idea to gamble at Fogelin's Palace.
What should you do: gamble at Fogelin's Palace or not? Why?