Exercises Probability and Statistics I
Introduction
You may find the answer to these exercises by drawing a Venn-diagram for the involved events.
1.
2.
A, B and C are events. Express the following events in A, B and C using union, intersection and
complement of these events (First draw a “general” Venn-diagram for A, B and C).
a. A and B, but not C. b. All three
c. At least out of three d. At least two
e. None
f. Exactly one of three g. Not more than 1
A, B and C are events. Express
in P(A), P(B), P(C), P(A B), P(B C), P(A C) and
P(A B C).
3.
4.
5.
We define for a collection of 1200 screws the following events:
A = the set of screws having a length of (at least) 10 cm.
B = the set of screws having a weight of (at least) 10 g.
C = the set of screws having a diameter of (at least) 10 mm.
400 screws have a length of 10 cm and a weight of 10 g.
400 screws have a length of 10 cm and a diameter of 10 mm.
400 screws have a diameter of 10 mm and a weight of 10 g.
300 screws have a length of 10 cm, a weight of 10 g and a diameter of 10 mm.
Compute the probability that an arbitrarily chosen screw is an element of at least 2 of the 3 events A, B , C.
We flip two coins at the same time. We could distinguish 3 outcomes of this experiment: twice Head, twice
Tail or one of both. D’Alembert stated in the 18 th Century that these outcomes are equally likely. Do you
agree? Why (not)?
a. Proof that
and
if
using the axioms of Kolmogorov.
b. Proof: P(A) ≤
6.
If P(A)= ½ ,
≤ P(A) + P(B)
= ⅓ and
, compute P(B) and
).
Combinatorial probability
1.
2.
3.
4.
5.
6.
What is the probability that an arbitrary 3-digit-number (chosen from 000 to 999) has 2 as lowest number?
A vase contains 3 red and 7 white balls. If we draw 4 balls at random (without replacement), what is the
probability that the fourth is red?
A deck of playing cards consists of 36 number cards (2-10 in 4 colours: spades, hearts, clubs and
diamonds)) and 16 picture-cards (ace, king, queen and jack, each in 4 colours). Suppose we draw 5 cards
arbitrarily from the deck (no replacement).
a. Compute the probability that all 5 are picture-cards.
b. Compute the probability that the first card is an ace and the last a king.
c. Compute the probability that among the 5 cards there is (exactly) one ace and one king.
Somebody claims to be a connoisseur of wine. To test his expertise he gets 6 cards with the names of 6
types of red wine. He is presented 6 glasses of these wines and, after tasting them, he has to put one card at
each glass. Suppose he is not an expert on wines at all.
a. What is the probability he will guess all wines right?
b. What is the probability that he will be acknowledged as a connoisseur of wine if you should have 4
names correct to be acknowledged.
A manufacturer of rubber rings guarantees that no more than 10% of the rubber rings is inferior. The rubber
rings are sold in packages of 100 rings. A buyer has a routine of testing each package: he chooses 10 rings
from the package, test them and if there is at least one inferior he will refuse the package.
a. What is the probability he refuses a package if it just satisfies the guarantee of the manufacturer?
b. What is the probability that the buyer accepts the package if 20% of the package is inferior?
We have put 10 balls, numbered 1 to 10, in a vase. We draw 4 balls arbitrarily without replacement.
What is the probability that the balls are drawn in ascending order (e.g. 2, 5, 7, 8)?
7.
A lottery consists of 100 lots. There are 4 first prices and 10 consolation prices. Prices are drawn without
replacements. Fatima bought 5 lots. What is the probability that she wins
a. one first and one consolation price b. exactly one price c. no price at all d. at least one price.
8. (Generalization of 7.) In a box there are N items: N1 of type 1, N2 of type 2, ...., Nk of type k, so ∑ Ni = N.
If n items are chosen from the box at random, what is the probability of n1 items of type 1, n2 of type 2, ....,
nk of type k, so that ∑ ni = n.
9. Alasana and Jamang divide 26 playing cards, of which 6 are spades, arbitrarily. Compute the probability
that a. one of both has exactly 4 of spades b. both Alasana and Jamang have 3 cards of spades
10. a. In how many ways can we divide 13 persons in two groups of 6 and 7 persons respectively.
b. In how many ways can we divide 30 persons in four groups of 6, 7, 8 and 9 persons respectively.
c. In how many ways can we divide n persons in k groups of n1 , n1 and nk persons respectively (∑ ni = n).
This number is called the multinomial coefficient (generalization of the binomial coefficient) .
Conditional probability and independence.
1.
2.
a. Compute
if
= ½ P(C) = 4/5 and
b. Compute
if
= 0.8,
= 0.01,
= 0.02 and
a. Compute the conditional probabilities P(A|B),
b. What does
4.
5.
6.
7.
8.
= 0.3 and
= 0.35.
Through a communication channel binary numbers (0 or 1) are sent. Because of noise on the line
sometimes a 0 is sent and a 1 is received or vice versa. A is the event that 1 is sent and B is the event that 1is
received.
are the events of sending and receiving 0.
Assume:
3.
=¾.
= 0.5
and
.
mean in words?
c. Compute the probability of a fault while sending a binary number.
30% of all families in a country has a television and 20% has a car nor a television. Furthermore we know
that is that 2/3 of all families with a television also has a car. Compute the probability that an arbitrary car
owner (family) has a television. (First define relevant events and express the given probabilities in terms of
conditional or unconditional probabilities involving these invents).
From a deck of (52) playing cards we draw 13 cards at random. Determine the probability that we have 3
cards of diamonds, 3 of hearts and 3 of diamond, provided that we have drawn 6 red (hearts or diamond)
and 7 black cards.
A cupboard has 3 drawers: in the first there are two golden coins, in the second one golden and one silver
coin and in the third two silver coins. Somebody opens an arbitrary drawer and chooses an arbitrary coin: it
is a golden coin. What is the probability that the second coin from this drawer is golden too?
In a vase there are 5 red and 7 white balls . We flip a dice and the number we flip is the number of draws
(without replacement) from the vase.
a. If we flip 5, what is the probability of 3 red balls?
b. Compute the probability that we draw 3 red balls. (First define events and express the given probability)
A car driver has caused an accident and has to take an alcohol test. Research has shown that 75% of the
persons who had (too much) alcohol shows a positive test result. However, 2% of the persons who did not
drink also show a positive test result. Assume that in case of accidents 5% of the car drivers have been
drinking alcohol. What is the probability that such a driver has a positive test result? (First define events!)
Prove (use a Venn diagram): a. A and B are independent => and B are independent (and also and )
b. A, B, C and D are independent events =>
9.
and CD are independent
We throw a dice nine times. Determine the probability that: a. the first three results are all 5 and the rest
different from 5 and
b. at most 2 throws result in 5 or 6.
10. Compute the probability that we need more than 6 throws with a dice to have the first 6 as a result.
11. A student is working at an exam with Multiple Choice Items. Each question has two possible answers (a, b),
one of which is correct. If he does not know the answer, he flips a coin (Tail = a, Head = b), Suppose that
he does not know the answer on 3/5th of the items.
a. What is the probability that he knew the answer to a question that he answered correct?
b. If there are 25 items, what is the probability of at least 23 correct answers?