Probability: Homework 1
Due:
1. (12%) De ne a sample space for the experiment of drawing two coins from a purse
that contains two quarters, three nickels, one dime, and four pennies. For the same
experiment describe the following events:
(a) drawing 26 cents;
(b) drawing more than 9 but less than 25 cents;
(c) drawing 29 cents.
2. (16%) Let E , F , and G be three events. Determine which of the following statements
are correct and which are incorrect. Justify your answers.
(a) (E − EF ) F = E F .
S S
(b) F c G E c G = G(F E)c .
S S
(c) (E F )c G = E c F c G.
S
(d) EF FG ⊂ E G.
S S S S
EG F
3. (8%) Prove De Morgan's second law, (AB)c = Ac B c , (a) by elementwise proof;
S
(b) by applying De Morgan's rst law to Ac and B c .
4. (5%) The probability that an earthquake will damage a certain structure during
a year is 0.015. The probability that a hurricane will damage the same structure
during a year is 0.025. If the probability that both an earthquake and a hurricane
will damage the structure during a year is 0.0073, what is the probability that next
year the structure will not be damaged by a hurricane or an earthquake?
5. (5%) Let A, B , and C be three events. Prove that
C) = P (A) + P (B) + P (C) − P (AB) − P (AC) − P (BC) + P (ABC).
S S
P (A B
6. (8%) A number is selected randomly from the set {1, 2, . . . , 1000}. What is the
probability that (a) it is divisible by 3 but not by 5; (b) it is divisible neither by 3
nor by 5?
7. (8%) Which of the following statements are true? If a statement is true, prove it.
If it is false, give a counterexample.
(a) If A is an event with probability 1, then A is the sample space.
(b) If B is an event with probability 0, then B = φ.
8. (5%) Suppose that a point is randomly selected from the interval (0, 1). Using the
de nition in Section 1.7, show that all numerals are equally likely to appear as the
rst digit of the decimal representation of the selected point.
9. (8%)
(a) Prove that − 1/2n, 1/2 + 1/2n) = {1/2}.
T∞
n=1 (1/2
(b) Using part (a), show that the probability of selecting 1/2 in a random selection
of a point from (0, 1) is 0.
10. (8%) In how many ways can we draw ve cards from an ordinary deck of 52 cards
(a) with replacement; (b) without replacement?
11. (5%) A number is selected randomly from the set {0000, 0001, 0002, . . . , 9999}.
What is the probability that the sum of the rst two digits of the number selected
is equal to the sum of its last two digits?
12. (12%)
(a) Prove that for any two events A and B , we have P (AB) ≥ P (A) + P (B) − 1.
(b) Generalize to the case of n events A1 , A2 , · · ·, An , by showing that
P (A1 A2 · · · An ) ≥ P (A1 ) + P (A2 ) + · · · + P (An ) − (n − 1).
(c) Let A1 , A2 , · · ·, An be n events. Show that if P (A1 )=P (A2 )=· · ·=P (An )=1,
then P (A1 A2 · · · An ) = 1.
Extra Exercises:
1. Let E , F , and G be three events; Explain the meaning of the relations E F
S S
G=
G and EF G = G.
2. Among 33 students in a class, 17 of them earned A's on the midterm exam, 14
earned A's on the nal exam, and 11 did not earn A's on either examination. What
is the probability that a randomly selected student from this class earned an A on
both exams?
3. From a small town 120 persons were selected at random and asked the following
question: Which of the three shampoos, A, B , or C , do you use? The following
results were obtained: 20 use A and C , 10 use A and B but not C , 15 use all three,
30 use only C , 35 use B but not C , 25 use B and C , and 10 use none of the three.
If a person is selected at random from this group, what is the probability that he
or she uses (a) only A; (b) only B ; (c) A and B ? (Draw a Venn diagram.)
4. A bus arrives at a station every day at a random time between 1:00 P.M. and 1:30
P.M. What is the probability that a person arriving at this station at 1:00 P.M. will
have to wait at least 10 minutes?
5. A delicatessen has advertised that it oers over 500 varieties of sandwiches. If at this
deli it is possible to have any combination of salami, turkey, bologna, corned beef,
ham, and cheese on French bread with the possible additions of lettuce, tomato, and
mayonnaise, is the deli's advertisement true? Assume that a sandwich necessarily
has bread and at least one type of meat or cheese.
