John claims that he is mildly psychic and can guess the color of a card withdrawn from an ordinary deck of cards correctly 70% of the time. His friend selects cards at random from the deck (replacing them and shuffling each time). He selects 12 cards and John guesses 9 correctly.
(a) Find the probability that John guesses exactly 9 correctly if his claim of being psychic is true.
(b) Find the probability that he gets exactly 9 correct if he is just guessing.

Question

John claims that he is mildly psychic and can guess the color of a card withdrawn from an ordinary deck of cards correctly 70% of the time. His friend selects cards at random from the deck (replacing them and shuffling each time). He selects 12 cards and John guesses 9 correctly. 
(a) Find the probability that John guesses exactly 9 correctly if his claim of being psychic is true. 
(b) Find the probability that he gets exactly 9 correct if he is just guessing.

Quizplus · Accepted Answer

The Answer is (a)

\left( \begin{array} { c } 12 \9\end{array} ight) ( .7 ) ^ { 9 } ( .3 ) ^ { 3 }

(b)

\left( \begin{array} { c } 12 \9\end{array} ight) ( .5 ) ^ { 9 } ( .5 ) ^ { 3 }

John Claims That He Is Mildly Psychic and Can Guess