Question

Suppose that we've decided to test Clara, who works at the Psychic Center, to see if she really has psychic abilities. While talking to her on the phone, we'll thoroughly shuffle a standard deck of 52 cards (which is made up of 13 hearts, 13 spades, 13 diamonds, and 13 clubs) and draw one card at random. We'll ask Clara to name the suit (heart, spade, diamond, or club) of the card we drew. After getting her guess, we'll return the card to the deck, thoroughly shuffle the deck, draw another card, and get her guess for the suit of this second card. We'll repeat this process until we've drawn a total of 18 cards and gotten her suit guesses for each. Assume that Clara is not clairvoyant, that is, assume that she randomly guesses on each card. Estimate the number of cards in the sample for which Clara correctly guesses the suit by giving the mean of the relevant distribution (that is, the expectation of the relevant random variable). Do not round your response. Quantify the uncertainty of your estimate by giving the standard deviation of the distribution. Round your response to at least three decimal places.

Answer 1

Answer : Suppose that we've decided to test Clara, who works at the Psychic Center, to see if she really has psychic abilities. While talking to her on the phone, we'll thoroughly shuffle a standard deck of 52 cards (which is made up of 13 hearts, 13 spades, 13 diamonds, and 13 clubs) and draw one card at random. We'll ask Clara to name the suit (heart, spade, diamond, or club) of the card we drew. After getting her guess, we'll return the card to the deck, thoroughly shume the deck, draw another card, and get her guess for the suit of this second card. We'll repeat this process until we've drawn a total of 18 cards and gotten her suit guesses for each. Assume that Clara is not clairvoyant, that is, assume that she randomly guesses on each card.

Solution:

a) Estimate the number of cards in the sample for which Clara correctly guesses the suit by giving the mean of the relevant distribution that is the expectation of the relevant random variable) :

There are total 18 cards drawn,

therefore, n = 18

Probability that she correctly guesses suit :

p = 13/52

= 0.25

Let x be the random variable that denotes the number of correct guesses by Clara.

Therefore, the expected value of the random variable x is:

E(x) = np

=18 * 0.25

= 4.5

Therefore,the number of cards in the sample for which Clara correctly guesses the suit by giving the mean of the relevant distribution is 4.5.

b) Quantify the uncertainty of your estimate by giving the standard deviation of the distribution :

The standard deviation :

S.D = √{np(1-p)}

= √{18 * 0.25 * (1-0.25)}

= 3.375

Therefore, the standard deviation of the distribution

is 3.375.