First year:
(I)
Introduction to probability
Probability: Introduction
Try to answer the following questions. Intuitively, you already know the
answer:
-
If you throw a dice, what is the probability of getting a 6 ?
-
If you throw a dice, what is the probability of getting a 5 or 6 ?
-
If you throw a dice, what is the probability of getting 1,2,3,4,5 or
6?
Definitions:
(1) The set S of all possible outcomes of a particular
experiment is called the sample space for the
experiment.
(2) An event is any collection of possible outcomes of an
experiment (i.e. any subset of S ).
In the example above, the sample space is S  1 , 2 , 3 , 4 , 5 , 6, the first event
is A  6 , the second event is B  5 , 6 and the third event is S itself.
General rule:
The probability of an event A is the number of possible occurrences of A
divided by the number of possible occurrences in the sample space S .
Formula: P ( A) 
#A
.
#S
Note that the number of possible occurrences in the sample space is always
greater than the number of possible occurrences in any given event A , so
that
0  P( A)  1 .
In the example above
#A 1
P( A) 
 ,
#S 6
P( B) 
Note that we always have P( S )  1 .
#B 2 1
  ,
#S 6 3
P( S ) 
#S 6
  1.
#S 6
(II)
Axioms of probability
Two events A and B are said to be disjoint if A  B   ,  being the empty
set.
For example, A  6 and B  5 , 6 are not disjoint since A  B  6.
On the other hand, B and C  2 , 4 are disjoint since B C  .
A1 , A2 ,... are said to be pairwise disjoint if Ai  A j   for i  j .
Axioms:
Given a sample space S and a set R of subsets of S , a probability function
is a function P from R into [0,1] satisfying
(i)
(ii)
(iii)
P( A)  0 for all A R ,
P( S )  1 ,
If A1 , A2 ,...  R are pairwise disjoint, then
P( A1  A2  ...)  P( A1 )  P( A2 )  ...
In our example, the event B  5 , 6 is the union of two disjoint events: 5 and
6. We have:
P( B)  P5 6  P5  P6 
1 1 1
  .
6 6 3
On the other hand, the last axiom does not hold if the events are not disjoint:
P( A  B)  P6 5 , 6  P5 , 6 
1
3
but
P( A)  P( B) 
1 1 1
  .
6 3 2
Exercise:
(1) In a group of 40 students, there are 22 females and 18 males. If one
student is chosen at random, what is the probability that the randomly
chosen student is
- a male,
- a female,
- either a male or a female?
(2) In a purse, there are 20 coins: four 2-euro coins, three 1-euro coins, six
50-cent coins and seven 20-cent coins. You pick a coin at random.
What is the probability the coin is
- a 20-cent coin.
- either a 20-cent coin or a 2-euro coin.
(3) You throw a dice twice (the first dice is, say, red and the second one is
yellow). What is the sample space here? What is the probability that
the sum of the face values is equal to 7? To 6? To 5? To 12?
Note: the probability of the sum of the face values being equal to 7 is 1/ 6 ,
while the probability of the sum of the face values being equal to 12 is
1/ 36 . In fact the event “sum of the face values = 7” is associated with the
highest probability, which explains why gamblers often bet on the sum
being equal to 7. On the other hand, even if you bet on 7, the probability of
loosing is still 83.33% , which explains why you shouldn’t gamble…
(III) More concepts and formulae
If two events A and B are not disjoint, i.e. if A  B   , then
P A  B  P( A)  P( B)  P A  B.
P( A  B) is the probability that the event A  B occurs or, equivalently, that
A and B occur at the same time.
In the first example, A  6 and B  5 , 6.
Then A  B  5 and P( A  B)  P( A)  P( B)  P( A  B) 
1 1 1 1
   .
6 3 6 3
You throw two dices once (the first dice is, say, red and the second one is
yellow). What is the probability of getting a 6 with the first dice and either a
six or a five with the second dice?
The two events are independent, meaning that one dice will not influence the
other.
If two events are independent, the probability that they occur simultaneously
1 1 1
is the product of the probabilities: P(6)  P5 , 6    .
6 3 18
On the other hand, the result of an experiment can influence the outcome of
another experiment. For example:
In a purse, there are 20 coins: four 2-euro coins, three 1-euro coins, six 50cent coins and seven 20-cent coins.
You pick a coin at random. Say you picked a 20-cent coin. You don’t put the
coin back in the purse and you pick another coin at random. What is the
probability that the second coin is a 20-cent coin?
You have picked a 20-cent coin the first time, so they are six 20-cent coins
left and 19 coins in the purse. Therefore the probability of picking a 20-cent
6
coin the second time is
.
19
Conditional probability
In a class of 100 fourth year students, 40 are twenty-five years old or more
and 30 of those 40 students are male.
A student is chosen at random. Let A be the event “the student is male” and
let B be the event “the student is twenty-five years old or more”.
What is the probability that the student is a male, knowing that the student
chosen at random is twenty-five years old or more?
This probability, called conditional probability, is denoted P( A | B ). Here, it is
fairly easy to see that
30 3
P( A | B) 
 .
40 4
More generally, we have for any conditional probability:
P( A  B)  P( B) P( A | B).
Let’s check that the formula holds (in our example):
The probability that a student is both male and twenty-five years old or more
is
30
3
P( A  B) 
 .
100 10
40
4
 , so
The probability that a student is 25 years old or more is P( B) 
100 10
that
P( A  B) 3 10 3
    P( A | B).
P( B)
10 4 4