You miss 100% of the shots...

13 Apr, 2024

You miss 100% of the shots you don't take. - Wayne Gretzky

Proof

Definitions

Let's define taking a shot as an event $S$, and hitting the shot as $H$.

Statement

$P(H\mid \neg S)=0$

Expansion

Mathematically, the probability of $H$ given $\neg S$ is represented by the conditional probability formula, though this specific scenario simplifies greatly:

$P(H\mid \neg S)=\frac{P(H\cap \neg S)}{P(\neg S)}$

Here, $P(H\cap \neg S)$ represents the probability of both $H$ and $\neg S$ occurring together. However, by the logical nature of the events:

• $H$ occurring (success or hitting the shot) necessitates taking the shot ($S$).
• If the shot isn't taken ($\neg S$), then hitting the shot ($H$) is impossible.

Thus, the intersection of $H$ and $\neg S$ (the probability of hitting the shot while not taking it) is 0: $P(H\cap \neg S)=0$

Assuming that the probability of not taking the shot is non-zero: $P(\neg S)\ne 0$

Therefore, using these values in the conditional probability formula gives: $P(H\mid \neg S)=\frac{0}{P(\neg S)}=0$

Conclusion

So, $P(H\mid \neg S)=0$ illustrates that it is impossible to achieve success in an event that one does not attempt. This mathematically validates the quote, "You miss 100% of the shots you don't take," since without attempting ($\neg S$), the outcome of success ($H$) cannot be achieved.

#failure #motivation #probability #success