Question

(14.18) In a test for ESP (extrasensory perception), a subject is told that cards the experimenter can see but he cannot contain either a star, a circle, a wave, or a square. As the experimenter looks at each of 5 cards in turn, the subject names the shape on the card. A subject who is just guessing has probability 0.25 of guessing correctly on each card.

Assume the subject's guesses are independent of each other.

The probability (±±0.0001) that the subject guesses the shape correctly on 2 out of the 5 cards is about

Answer 1

As there are 4 shapes, each time the probability of correct guess is computed as: 1/4 = 0.25
Therefore out of 5 cards, the number of times the correct guess is made could be modelled here as a binomial distribution such that:

$X \sim Bin(n= 5, p = 0.25)$
Now the required probability could be computed using the binomial probability function as:

$P(X = 2) = \binom{5}{2}0.25^2 * (1 - 0.25)^3 = 10*0.25^2*0.75^3 = 0.2637$

Therefore 0.2637 is the required probability here.