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Suppose that 8% of all bicycle racers use steroids. A bicyclist who uses steroids tests positive for steroids 96% of the time, and that a bicyclist who does not use steroids tests positive for steroids 9% of the time. What is the probability that a randomly selected bicyclist who tests positive for steroids actually uses steroids?

A) 11.23% B) 34.30% C) 48.12% D) 56.13%

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Bicyclists who test actual positive (who actually use steroids) = 0.08 * 0.96 = 0.0768

Now, bicyclists who test false positive (i.e. the ones who do not use steroids but still test positive) = 0.92 * 0.09 = 0.0819

$\therefore$ Total bicyclists who test positive = 0.0768 + 0.0828 = 0.1596

Favorable cases (the ones who actually use steroids) = 0.0768

So, required probability = $\frac{0.0768}{0.1596}$ = 0.4812 = 48.12%

Now, bicyclists who test false positive (i.e. the ones who do not use steroids but still test positive) = 0.92 * 0.09 = 0.0819

$\therefore$ Total bicyclists who test positive = 0.0768 + 0.0828 = 0.1596

Favorable cases (the ones who actually use steroids) = 0.0768

So, required probability = $\frac{0.0768}{0.1596}$ = 0.4812 = 48.12%

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