Psychology 406 Assignment #2

1) Hays — Chapter 1, #18; Chapter 2, #20; Chapter 3, #18, #20

2) Suppose a fair coin is flipped four times and the number of heads, X, is recorded. Assume that each different sequence of heads and tails has the same probability of occurring, e.g., P{(H,T,H,T)} = 1/16.

a) Complete the theoretical probability distribution for X: P(X = 0) = P(X = 1) = P(X = 2) = P(X = 3) = P(X = 4) =

b) Graph both the probability mass function and the cumulative distribution function for X.

c) Calculate:

i) P(1 ≤ X ≤ 3) = ii) P(X ≥ 4) = iii) P(X < 1) = iv) P(1 < X < 2) =

3) Now, actually flip a coin four times (or four coins once) for twenty different trials, and record the number of heads in a frequency table.

a) Draw a histogram, frequency polygon, and a cumulative frequency distribution for your results.

b) Do your results conform to the theoretical probability distribution? Should they? What would you expect to obtain in twenty trials based on your answer in 2(a)?

4) Consider a pointer calibrated as in the diagram below:

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1; 2; 3; 4; 5]

a) Draw the probability density function and the cumulative distribution function for X = spot where the pointer lands.

b) Using the probability density function as a guide, find the following probabilities:

P(1 < X < 2) = P(1 ≤ X ≤ 2) = P(2 ≤ X) = P(X ≤ 3) =

c) Represent algebraically each of the four probabilities above in terms of the cumulative distribution function, F(a) = P(X ≤ a).

5) Suppose we draw a card from a normal deck and toss a 6-sided die.

a) What would be the sample space? How many members?

b) Assume all members of the sample space are equally likely. Let A be the event of obtaining a Queen; B the event of obtaining a 4 on the die.

How many members do the sets A and B contain separately? What is P(A), P(B)? How many members does A∩B contain? What is P(A∩B)? Are A and B independent?

c) In what way is the assumption of equally-likely elements of the sample space important?

6) Suppose I have a “fair” 4-sided die, i.e., a pyramid with either 1, 2, 3, or 4 spots on a side, where the probability of any one side turning face down is 1/4. The outcome associated with any one trial will be the number of spots on the side of the die that is facing down.

a) If I toss the die four times and record the outcomes sequentially, e.g., (1,3,2,2), how many different possible outcomes could be obtained?

How many ways could I observe a sequence that did not contain a repeat of numbers, e.g., (1,3,2,4), (3,1,2,4), etc.

b) If I toss the die three times instead of four, how many ways could I observe a sequence that did not contain a repeat of numbers, e.g., (1,4,3), (1,3,2), etc.

Now, if I did not care about the order of the sequences containing no repeats, how many different outcomes could I observe? Put your answer in terms of a binomial coefficient.

c) Assume that the die is “loaded”, i.e., the four outcomes of a toss are not equally-likely. Let X = 1, 2, 3, or 4 depending on the outcome of a single toss, and suppose the probabilities are as follows:

P(X = 1) = 3/4

P(X = 2) = 1/8

P(X = 3) = 1/8

P(X = 4) = 0

If the die is tossed four times and the four trials do not affect one another, what is the probability of observing the following sequences?

i) (1,1,1,1)

ii) (1,4,3,1)

iii) (1,2,3,3)

If the die is tossed four times and Y = number of 1’s in the four trials, complete the theoretical probability distribution for Y. (Hint: How is Y related to a binomial random variable?)

7) Suppose we pick a person at random and force a Republican or a Democrat disclosure (only two acceptable labels). If the probability of responding “Democrat” is 3/4, write expressions for the following (do not actually compute):

a) In a random sample of 1,000 people, the probability that 200 will say “Democrat” and 800 will say “Republican”.

b) In a random sample of 1,000 people, the probability that 50 or less will say “Democrat”.

c) In a random sample of 1,000 people, the probability that 400 to 600 people will say “Republican”.