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.