Imagine a population divided into four groups:

1. Male, Gamer
2. Female, Gamer
3. Male, Non-Gamer
4. Female, Non-Gamer

We want to know: if a randomly chosen person is male, what is the probability that they are an Gamer?

1. By definition,

   $$ P(\text{Male}\mid \text{Gamer}) = \frac{\text{number of Male Gamers}}{\text{number of Gamers}}. $$

2. Similarly,

   $$ P(\text{Gamer}\mid \text{Male}) = \frac{\text{number of Male Gamers}}{\text{number of Males}}. $$

3. Since

   $$ \text{number of Male Gamers} = P(\text{Gamer}\mid \text{Male}) \times \text{number of Males}, $$

   we can write

   $$ P(\text{Male}\mid \text{Gamer}) = \frac{P(\text{Gamer}\mid \text{Male})\\;\times\\;\text{number of Males}} {\text{number of Gamers}}. $$

4. Dividing numerator and denominator by the total population size $N$, we get

   $$ P(\text{Male}\mid \text{Gamer}) = \frac{P(\text{Gamer}\mid \text{Male})\\;\times\\;\dfrac{\text{number of Males}}{N}} {\dfrac{\text{number of Gamers}}{N}} = \frac{P(\text{Gamer}\mid \text{Male})\\;P(\text{Male})}{P(\text{Gamer})}. $$

This is Bayes’ theorem in its classic form:

$$ \boxed{ P(\mathrm{Male}\mid \mathrm{Gamer}) = \frac{ \overset{\text{likelihood}}{P(\mathrm{Gamer}\mid \mathrm{Male})} \\;\times\\; \overset{\text{prior}}{P(\mathrm{Male})} }{ \overset{\text{evidence}}{P(\mathrm{Gamer})} } } $$

• Prior $P(\text{Male})$: your initial belief about how likely someone is an Male.
• Likelihood $P(\text{Gamer}\mid \text{Male})$: how probable it is to observe “Gamer” among Males.
• Evidence $P(\text{Gamer})$: the overall chance of picking a Gamer.
• Posterior $P(\text{Male}\mid \text{Gamer})$: your updated belief in “Male” once you know “Gamer.”

With Numbers

Here’s the same survey broken down again:

	Gamer	Non‑Gamer	Total
Male	30	70	100
Female	50	50	100
Total	80	120	200

We now want

$$ P(\text{Male}\mid \text{Gamer}). $$

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1. Direct calculation

By definition of conditional probability,

$$ P(\text{Male}\mid \text{Gamer}) = \frac{\text{Male \\& Gamer}}{\text{Gamer}} = \frac{30}{80} = 0.375 $$

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2. Via Bayes’ theorem

Bayes says

$$ P(A\mid B) = \frac{P(B\mid A)\,P(A)}{P(B)}. $$

Let

• $A={\text{Male}}$,
• $B={\text{Gamer}}$.

So

$$ P(\text{Male}\mid \text{Gamer}) = \frac{P(\text{Gamer}\mid \text{Male})\,P(\text{Male})}{P(\text{Gamer})}. $$

We already know:

• $P(\text{Gamer}\mid \text{Male}) = \dfrac{30}{100} = 0.30.$
• $P(\text{Male}) = \dfrac{100}{200} = 0.50.$

2.1. Computing the evidence $P(\text{Gamer})$

Using the law of total probability over Male/Female:

$$ P(\text{Gamer}) = P(\text{Gamer}\mid \text{Male})\,P(\text{Male}) +P(\text{Gamer}\mid \text{Female})\,P(\text{Female}). $$

We have

• $P(\text{Gamer}\mid \text{Female}) = \dfrac{50}{100} = 0.50,$
• $P(\text{Female}) = 0.50.$

Hence

$$ P(\text{Gamer}) = 0.30\times0.50 + 0.50\times0.50 = 0.15 + 0.25 = 0.40. $$

2.2. Plug in and finish

$$ P(\text{Male}\mid \text{Gamer}) = \frac{0.30 \times 0.50}{0.40} = \frac{0.15}{0.40} = 0.375. $$

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3. Conclusion

Whether by direct counting or by Bayes’ theorem (with the evidence computed via male/female breakdown), we get

$$ \boxed{P(\text{Male}\mid \text{Gamer}) = 0.375.} $$

Why it matters

Bayes’ rule tells you exactly how to update your prior belief in light of new evidence. In this example:

$$ \text{posterior} \\;\longleftarrow\\; \frac{\text{likelihood}\times\text{prior}}{\text{evidence}}. $$

The more informative your likelihood, the more your posterior shifts away from the prior.