NAME:

To test whether a person has telepathy (the ability to read minds), researchers use a special deck of cards that have five different shapes: Star, Diamond, Circle, Wavy Lines, and Square. Cards are drawn at random, with replacement, from a shuffled deck. The "sender" selects a card and concentrates, and the "receiver" (who is usually in another room and can't see the sender) says what he or she thinks the card is.

Suppose 100 cards are drawn and "sent". Let X represent the number of correct answers by the receiver.

1. What is the probability density of X, and why? (i.e. what are the conditions required for this to be the pdf.)

 X is a binomial random variable, with n = 100, p = .20 (or 1/5). For this to be true, there have to be a finite number of independent trials (which is the case as long as the result of one draw of the cards can provide no information about future draws -- and this will be true if the receiver doesn't know of the results and particularly if the deck has a very large number of well-shuffled cards.) The outcome has to be binary: yes or no, and in this case the outcome is "success" or "failure". The random variable has to be the number of successes, which it is. Also, the probability of success has to be the same at every trial (which it is under the previous assumptions).

2. What are the mean and SD of X?

 The mean of a binomial random variable is np = 100*.20 = 20. The SD is sqrt(np(1-p)) = sqrt(100*.2*.8) = 4.

3. Approximately what percentage of the population will get more than 28 correct, assuming that no one in the population has telepathy?