## 32. Xenon, Zephyr, and Enofa

Two friends at the races were speculating about the outcome of the next race. One of them bet on Xenon and Enofa, while the other bet on Zephyr.

- I bet,- said the second,- that if my horse finishes in the top three, then Xenon will also be in the top three.
- And I,- replied the first,- bet that if at least one of my horses finishes in the top three, then you will lose your bet.

Suppose the last bet was not lost. Which of the three horses, Xenon, Zephyr, or Enofa, in this case, has the highest chance of finishing in the top three?
32. Let $X, Y$ and $Z$ denote the following three statements:

$X$ - Xenon is in the top three;

$Y$ - Enofa is in the top three;

$Z$ - Zephyr is in the top three.

Since the second bet was not lost, if one of the statements, $X$ or $Y$, turned out to be true, then the first bet was lost. But if the first bet was lost, then both $Z$ and not $X$ (the negation of $X$) were true; we will denote it as $\bar{X}$. Thus, we see that if $X$ or $Y$ were true, then $Z$ and $\bar{X}$ were also true. From this, it follows that $X$ could not have been true. Moreover, it is known that if $Y$ is true, then $Z$ is also true, i.e., Zephyr has at least as good a chance of making it into the top three as Enofa, while Xenon has no chance at all. Therefore, the most promising horse is Zephyr.