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

16

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.

(If necessary, consult a list of formulas.)

Answer 1

Hello Sir/ Mam

There are only 2 choices/ happenings when Clara guesses the suit of the card drawn, either she guesses correctly or not.

Hence, we can say that the random variable follows binomial distribution.(as card is replaced after the draw)

Now, as there are 4 suits and clara has to select 1, hence, lets call guessing correctly, success. Then p = 0.25.

Now, when n = 16,

$X\sim Bin(16,0.25)$

(a) The number of cards in the sample for which Clara correctly guesses the suit= Mean = 4

$\mu = np = 16*0.25=4$

(b)  Uncertainty of the estimate = Standard Deviation = 1.7321

$\sigma = \sqrt{np(1-p) }= \sqrt{16*0.25*0.75}=1.7321$

I hope this solves your doubt.

Feel free to comment if you still have any query or need something else. I'll help asap.

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