Question

In a test for extrasensory perception, the experimenter looks at cards containing either a star, circle, wave, or square. (The subject cannot see the cards.) As the experimenter looks at each of 20 cards, the subject names the shape on the card. Assuming that any success guessing shapes is due purely to chance, what is the probability a subject correctly guesses at least 10 of the 20 shapes? (Round to 3 decimal places)

Answer 1

Answer:

The probability of a subject correctly guesses at least 10 of the 20 shapes is 0.014

Step-by-step explanation:

Looking at the experiment, we can find the probability thinking it as a Bernoulli experiment (dicotomic variable), as the subject either guess the shape right or not, and each guess is independent from the others (as are due to chance).

We are going to use a binomial distribution (useful to model successes in n   Bernoulli experiments), as we want to know the probability of at least 10 right guesses (number of sucess) in 20 cards (number of independent experiments). The probability of guessing right a shape is:

$\mbox{Probability of guessing right}=\frac{\mbox{favorable cases}}{\mbox{total cases}}=\frac{1}{4}$

The probability of k sucesses in n experiments, each with a p probability of sucess is given by:

$P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$

As we want to know the probability of getting at least 10 correct guesses, we have to add the odds for 10, 11, 12, ..., 20 right guesses:

$P(X \ge 10) = \sum_{i=10}^{20} {20\choose i}0.25^i(1-0.25)^{20-i}=0.014$