# Bayesian inference and a priori probability

A woman claims she has "magical powers" and can guess the correct value of any card in a four-card deck 50% of the time (of course, for someone without magical powers, this value would be only 25% of the time).

To test this claim, the woman guesses the value of a card from a shuffled deck six times. She guesses the correct value of a card 4 out of the 6 times.

How high must the a priori probability be that the woman has magical powers in order that the a posteriori probability that she has magical powers is 0.75?

How should I go about solving this? Is there a formula that I should be using? Should I use Bayes' Theorem in some way? I'm lost and any hints to get me started would be extremely useful.

Thank you!

• Something to consider - what do you need to apply Bayes theorem? What is "the data"? – probabilityislogic Jan 30 '17 at 3:32
• @probabilityislogic Well, if I plug the values into Bayes' Theorem, I get $P(\text{magical powers | correct guess})=\displaystyle\frac{P(\text{correct guess | magical powers}) \cdot P(\text{magical powers})}{P(\text{correct guess})} = \displaystyle\frac{0.5 \cdot 0.75}{0.25^2 + (0.75 \cdot 0.5)}\approx0.857$. Is this the correct use of the data? If so, where does the idea of a priori and a posteriori come in? – Bing Jan 30 '17 at 4:02
• How have you incorporated the fact that $4$ out of $6$ are guessed correctly? – probabilityislogic Jan 30 '17 at 4:26
• @probabilityislogic Okay, so would I calculate $P(\text{magical powers | 4 out of 6 correct guesses})$ instead? What would I do with that probability once I calculated it, though? – Bing Jan 30 '17 at 4:30
• You solve for $p (\text {magical powers})$ as the question asks. – probabilityislogic Jan 30 '17 at 5:42