Singing Competition

Ana, Becky and Claire are three singers entering a contest. Ana has won 4% of past competitions, Becky has 5%, and Claire has 6%. If Ana has submitted 7 albums, Becky 2 and Claire 3, what is the probability that Ana will win this time?

The general formula of Bayes’ theorem is:

$\\ P(j|W) = \frac{P(W|j)*P(j)}{\sum\limits_{i = 1}^{n} P(W|i)*P(i)} \\ \\ \frac{P(W|j)*P(j)}{P(W|1)*P(1) + P(W|2)*P(2) + P(W|3)*P(3)}$

In the present case, for Ana, it is:

$\\ P(Ana|W) = \frac{P(W|Ana)*P(Ana)}{P(W|Ana)*P(Ana) + P(W|Becky)*P(Becky) + P(W|Claire)*P(Claire)} \\\\ \frac{0.04 *(7/12)}{0.04 *(7/12) + 0.05 *(2/12) + 0.06 *(3/12)} = 0.5$

50% chance!

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