A sportsperson could be an expert in badminton and / or squash. Given that someone is a squash expert, the probability that they are expert at badminton as well is 0.8. There are twice as many badminton experts as there are squash experts. Given that Saina is a badminton expert, what is the likelihood that she is an expert in squash?

Let's use B for badminton expert and S for squash expert.

What we know:

Information	Mathematical form
If someone is a squash expert, probability they're also a badminton expert is 0.8	$P(B \mid S) = 0.8$
There are twice as many badminton experts as squash experts	$P(B) = 2 \times P(S)$

What we need to find:

Given Saina is a badminton expert, what's the probability she's also a squash expert? This is $P(S \mid B)$.

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We'll use Bayes' Theorem, which connects these probabilities:

$P(S \mid B) = \frac{P(B \mid S) \times P(S)}{P(B)}$

This formula helps us "flip" a conditional probability. We know $P(B \mid S)$ but need $P(S \mid B)$.

Substituting what we know:

$P(S \mid B) = \frac{0.8 \times P(S)}{2 \times P(S)}$

The $P(S)$ cancels out from numerator and denominator:

$P(S \mid B) = \frac{0.8}{2} = 0.4$

Why does this make sense?

Among squash experts, 80% also play badminton (high overlap). But there are twice as many badminton experts overall, meaning many badminton experts don't play squash. So the reverse probability is lower: only 40% of badminton experts also play squash.

Answer: 0.4