Lab #4
Answer the following questions on a SEPARATE sheet of paper.
1. A couple plans to have three children. There are 8 possible arrangements of girls and boys. Assume that all 8 outcomes are equally likely.
a) What is the sample space for this situation?
b) What is the probability for each of these outcomes?
c) Suppose we now define a random variable X to be the number of girls that the couple has. What are the possible values for X?
d) What is the probability for each of these values?
2. Choose an American household (that owns no more than 5 vehicles) at random and let the random variable X be the number of vehicles they own. Here is the probability model for these households:
Value of X / 0 / 1 / 2 / 3 / 4 / 5Probability / 0.09 / 0.36 / 0.35 / 0.13 / 0.05 / 0.02
a) Verify that this is a legitimate discrete distribution. (Note: You need to check two things.)
b) Say in words what the event {X ≥ 1} is. Find P(X ≥ 1).
c) A housing company builds houses with two-car garages. What percent of households have more cars than the garage can hold?
3. Dragons have been terrorizing a village. Suppose that 47% of them breathe fire and have bad breath, while 98% have at least one of these bad habits. Suppose too that 38% of them only breathe fire.
a) Let A be the event that a dragon breathes fire. Let B be the event that the dragon has bad breath. Using the above information, construct a probability table. A probability table looks like the following: ( Notes: 1. The ∩ symbol means “and”, i.e. P(A ∩ B) = P(A and B).
2. AC means “not A”, so
P(A ∩ BC) = the probability that event A occurs and event B does not. )
A
/ ACB
/ P(A B) / P(AC B)BC / P(A BC) / P(AC BC)
Notice that these tables are constructed in such a way that the four boxes are disjoint!
You need to determine the values for
P(A ∩ B), P(AC ∩ B), etc.
b) What is the probability that a dragon has only bad breath?
c) Are the events “breathing fire” and “bad breath” independent? Why or why not?
( Note: Two events, A and B, are independent if P(A ∩ B) = P(A) P(B). )
4. The table below shows the approximate U.S. age distribution for the year 2005.
Population / 28% / 20% / 35% / 15% / 2%
a) What is the probability that a randomly selected person in the United States will be at least
20 years old?
b) What is the probability that a randomly selected person in the United States will be less
than 60 years old?
5. The histogram below shows the distribution of hurricanes that have hit the U.S. mainland by
Category, with 1 the weakest level and 5 the strongest. Use the histogram to find the
a) mean
b) variance
c) standard deviation
d) expected value of the probability distribution, and
e) interpret the results