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Probability
Outcome :
An outcome is the result of an experiment or other situation involving uncertainty. The set of
all possible outcomes of a probability experiment is called a sample space.
Sample Space :
The sample space is an exhaustive list of all the possible outcomes of an experiment. Each
possible result of such a study is represented by one and only one point in the sample space,
which is usually denoted by S.
Examples
Experiment Rolling a die once:
Sample space S = {1,2,3,4,5,6}
Experiment Tossing a coin:
Sample space S = {Heads,Tails}
Experiment Measuring the height (cms) of a girl on her first day at school:
Sample space S = the set of all possible real numbers
Event :
An event is any collection of outcomes of an experiment. Formally, any subset of the sample
space is an event.
Any event which consists of a single outcome in the sample space is called an elementary or
simple event. Events which consist of more than one outcome are called compound events.
Set theory is used to represent relationships among events. In general, if A and B are two
events in the sample space S, then
(A union B) = 'either A or B occurs or both occur'
(A intersection B) = 'both A and B occur'
(A is a subset of B) = 'if A occurs, so does B'
A' or or Ac = 'event A does not occur'
(the empty set) = an impossible event
S (the sample space) = an event that is certain to occur
Example
Experiment: rolling a dice once Sample space S = {1,2,3,4,5,6}
Event A = 'score < 4' = {1,2,3}
B = 'score is even' = {2,4,6}
C = 'score is 7' =
= 'the score is < 4 or even or both' = {1,2,3,4,6}
= 'the score is < 4 and even' = {2}
A' or = 'event A does not occur' = {4,5,6}
Probability
The theory of Probability was developed with the study of games of chance such as
playing with dice and cards. Along with games we also face uncertainty in different phases of
life like business, projects, economy and in many other day to day activities of life. In all
these areas we face uncertainty and so probability plays a great role in these areas of life.
A probability provides a quantitative description of the likely occurrence of a particular
event. Probability is conventionally expressed on a scale from 0 to 1; a rare event has a
probability close to 0, a very common event has a probability close to 1.
In some experiments, all outcomes are equally likely. Suppose in an experiment there
are n equally likely outcomes and m outcomes corresponds to event E. This is the equallylikely outcomes model and is defined to be:
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number of outcomes corresponding to event E
P(E) =
total number of outcomes
Examples
1. The probability of drawing a spade from a pack of 52 well-shuffled playing cards is
13/52 = 1/4 = 0.25 since
event E = 'a spade is drawn';
the number of outcomes corresponding to E = 13 (spades);
the total number of outcomes = 52 (cards).
2. When tossing a coin, we assume that the results 'heads' or 'tails' each have equal
probabilities of 0.5.
Subjective Probability
A subjective probability describes an individual's personal judgement about how likely a
particular event is to occur. It is not based on any precise computation but is often a
reasonable assessment by a knowledgeable person.
Like all probabilities, a subjective probability is conventionally expressed on a scale from 0 to
1; a rare event has a subjective probability close to 0, a very common event has a subjective
probability close to 1.
A person's subjective probability of an event describes his/her degree of belief in the event.
Example
A Indian supporter might say, "I believe that Indians have probability of 0.9 of winning the
Cricket Cup since they have been playing really well."
Different types of events in probability:
1. Sure event: The event which has the probability of 1 i.e. 100 %, then it is a Sure Event. In
this type of event we surely get the output in every sample experiment is performed.
Ex.: Tossing a coin : event of getting head or tail.
2. Impossible Event: An event is called Impossible Event when we find that the experiment
checked never occurs. It means the probability of such event is 0%.
Ex.: Throwing a die : event of getting number more than 6.
3. Simple event : An event that consists of exactly one outcome.
4. Compound event: An event is called compound event, when if is the disjoint Union of two
or more Elementary Events. In other words, an event having more than one sample Point is
called a compound event. We can also say that event is a compound event which is a mixed
event and which is not a simple event.
Examples of simple and compound Event :
If a single face is considered when a die is rolled, then it will be simple event. For
example suppose getting 5 or 6 or 3 or 2 etc… on the die when it is thrown, is called as
simple event. If the event is any even number on the die, then the event is consist of points
{2, 4, 6}, which is known as compound event. That compound event is consisting of three
simple events i.e., {2}, {4} and {6}.
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1. Suppose two dice are rolled simultaneously, then the pair (1, 1) will be the simple
event. This is so, because it is a single outcome in the sample space. If event consists
of the sum of two dice is ‘’5’’ than it consists of four outcomes i.e., (1, 4), (2, 3), (3, 2),
(4, 1) and this is considered to be a compound event.
2. Suppose two coins are tossed simultaneously, then the pair (HT) will be the simple
event. If condition is defined that an event should consist of at least one head then
there are three outcomes. These outcomes are (HH), (HT) and (TH) and this is said to
be as compound event. That compound event consists of three simple events i.e.,
{HH}, {HT} and {TH}.
5. Complementary Event : Let A be the event, the complementary event to A is the event
where A does not occur. It is denoted by Ac or A’ or Ā.
5. Exclusive events: Two or more events are called exclusive when we try the experiment
and find that no one of them can occur at the same time, which means the possibility of only
one event among the two is possible at a time.
Ex.: Getting head or tail are exclusive events.
6. Exhaustive events: A set of events, where the union of all the experiments in the space is
the complete space itself, is called Exhaustive Events.
7. Dependent Events : The events which means they can be affected by previous events.
Example: Drawing 2 Cards from a Deck
After taking one card from the deck there are less cards available, so the probabilities
change!
Let's say you are interested in the chances of getting a King.
For the 1st card the chance of drawing a King is 4 out of 52
But for the 2nd card:
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
If the 1st card was a King, then the 2nd card is less likely to be a King, as only 3 of
the 51 cards left are Kings.
If the 1st card was not a King, then the 2nd card is slightly more likely to be a King, as
4 of the 51 cards left are King.
8. Disjoint Events
Let A and B be events in a probability space.
Then A and B are disjoint iff ::
A∩B=∅
It follows by definition of probability measure that A and B are disjoint iff:
Pr(A∩B)=0
That is, two events are disjoint iff the probability of them both occurring in the same
experiment is zero.
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That is, iff A and B can't happen together.
A and B are also referred to in this context as mutually exclusive.
Mutually Exclusive Events :
Two events are mutually exclusive (or disjoint) if it is impossible for them to occur
together.
Formally, two events A and B are mutually exclusive if and only if
If two events are mutually exclusive, they cannot be independent and vice versa.
Examples
1. Experiment: Rolling a die once
Sample space S = {1,2,3,4,5,6}
Events A = 'observe an odd number' = {1,3,5}
B = 'observe an even number' = {2,4,6}
= the empty set, so A and B are mutually exclusive.
2. A subject in a study cannot be both male and female, nor can they be aged 20 and 30.
A subject could however be both male and 20, or both female and 30.
Independent Events
Two events are independent if the occurrence of one of the events gives us no
information about whether or not the other event will occur; that is, the events have no
influence on each other.
In probability theory we say that two events, A and B, are independent if the probability that
they both occur is equal to the product of the probabilities of the two individual events, i.e.
The idea of independence can be extended to more than two events. For example, A, B and
C are independent if:
a. A and B are independent; A and C are independent and B and C are independent
(pairwise independence);
b.
If two events are independent then they cannot be mutually exclusive (disjoint) and vice
versa.
Example
Suppose that a man and a woman each have a pack of 52 playing cards. Each draws a card
from his/her pack. Find the probability that they each draw the ace of clubs.
We define the events:
A = probability that man draws ace of clubs = 1/52
B = probability that woman draws ace of clubs = 1/52
Clearly events A and B are independent so:
= 1/52 . 1/52 = 0.00037
That is, there is a very small chance that the man and the woman will both draw the ace of
clubs.
Addition Rule
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The addition rule is a result used to determine the probability that event A or event B occurs
or both occur.
The result is often written as follows, using set notation:
where:
P(A) = probability that event A occurs
P(B) = probability that event B occurs
= probability that event A or event B occurs
= probability that event A and event B both occur
For mutually exclusive events, that is events which cannot occur together:
=0
The addition rule therefore reduces to
= P(A) + P(B)
For independent events, that is events which have no influence on each other:
The addition rule therefore reduces to
Example
Suppose we wish to find the probability of drawing either a king or a spade in a single draw
from a pack of 52 playing cards.
We define the events A = 'draw a king' and B = 'draw a spade'
Since there are 4 kings in the pack and 13 spades, but 1 card is both a king and a spade, we
have:
= 4/52 + 13/52 - 1/52 = 16/52
So, the probability of drawing either a king or a spade is 16/52 (= 4/13).
Multiplication Rule
The multiplication rule is a result used to determine the probability that two events, A and B,
both occur. The multiplication rule follows from the definition of conditional probability.
The result is often written as follows, using set notation:
where:
P(A) = probability that event A occurs
P(B) = probability that event B occurs
= probability that event A and event B occur
P(A | B) = the conditional probability that event A occurs given that event B has
occurred already
P(B | A) = the conditional probability that event B occurs given that event A has
occurred already
For independent events, that is events which have no influence on one another, the rule
simplifies to:
That is, the probability of the joint events A and B is equal to the product of the individual
probabilities for the two events.
Conditional Probability
In many situations, once more information becomes available, we are able to revise
our estimates for the probability of further outcomes or events happening. For example,
suppose you go to shopping at the same place and time every Sunday and you finish
shopping within 15 minutes with probability 0.85. However, given that the festival season the
shop is exceptionally busy, the probability of finishing shopping within 15 minutes may
reduce to 0.7. This is the conditional probability of shopping within 15 minutes given that the
shop is exceptionally busy.
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The usual notation for "event A occurs given that event B has occurred" is "A | B" (A
given B). The symbol | is a vertical line and does not imply division. P(A | B) denotes the
probability that event A will occur given that event B has occurred already.
A rule that can be used to determine a conditional probability from unconditional probabilities
is:
where:
P(A | B) = the (conditional) probability that event A will occur given that event B has
occured already
= the (unconditional) probability that event A and event B both occur
P(B) = the (unconditional) probability that event B occurs
Law of Total Probability
The result is often written as follows, using set notation:
where:
P(A) = probability that event A occurs
= probability that event A and event B both occur
= probability that event A and event B' both occur, i.e. A occurs and B does
not.
Using the multiplication rule, this can be expressed as
P(A) = P(A | B).P(B) + P(A | B').P(B')
Bayes' Theorem
Bayes' Theorem is a result that allows new information to be used to update the conditional
probability of an event.
Using the multiplication rule, gives Bayes' Theorem in its simplest form:
Using the Law of Total Probability:
P(B | A).P(A)
P(A | B) =
P(B | A).P(A) + P(B | A').P(A')
where:
P(A) = probability that event A occurs
P(B) = probability that event B occurs
P(A') = probability that event A does not occur
P(A | B) = probability that event A occurs given that event B has occurred already
P(B | A) = probability that event B occurs given that event A has occurred already
P(B | A') = probability that event B occurs given that event A has not occurred already.
Examples :
Genes on the bacterial chromosome have the following linkages in conjugal transfer: x
and y, 25% of the time; y and z, 50% of the time. If the gene order is x-y-z, approximately
what percentage of the time will x and z be transferred together?
Chance that a Y will come with a Z (.5) * chance that an X will come with that Y (.25)
.25 * .50 = .125 ≈ 13%
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In a group of mice there are 200 white mice and 50 brown mice:
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

The proportional frequency of brown mice is 50/250 = 1/5 = 0.2
If we randomly take one mouse there is a 1/5 chance of it being brown (20%).
The probability of picking a brown mouse as a single random sample is equivalent to
the proportional frequency of brown mice in the group / population.
If there were 250 white mice, the probability of selecting a brown mouse would be 0/250 =
0.
The probability of selecting a white mouse would be 250/250 = 1 (100%).

P is normally written as a decimal, e.g. P = 0.5 - all probabilities lie between 0 and 1
(0% and 100%).
Applications of probability science in Genetics(some examples).
The cells in your body receive half their chromosomes from your father and one half of
their chromosomes from your mother. So for each pair of homologous chromosomes one will
be a maternal chromosome and one will be a paternal chromosome.. During meiosis when
the haploid gametes are formed, each member of the pair but not both ends up in a gamete.
This is the principle of segregation.
1. How many possible combinations of maternal and paternal chromosomes are there?
Answer. For any single pair of chromosomes either the gamete has a maternal chromosome
or a paternal chromosome, so there are two possibilities for any single pair of chromosomes.
Since we have 23 pairs of homologous chromosomes there are 2^23 possible combinations
of maternal Vs paternal chromosomes in the gametes.
2. How many of these combinations involve 10 maternal chromosomes and 13 paternal
chromosomes?
Answer: We want the total number of ordered combinations involving 10 maternal and 13
paternal chromosomes. So use the formula:
N!/[(M!)(N-M)!] where N = 23 and M = 13
3. What is the probability of getting a gamete with 10 maternal and 13 paternal
chromosomes? Assume any ordered combination is equally likely.
4. Recall that DNA is made from a sequence of four nucleotides that differ in terms of which
nitrogen base A,T,G,C is present in each nucleotide.
Suppose you have a region of DNA which is 100 nucleotides long. How many possible
nucleotide sequences are there for this region of DNA?
5. For maternal and paternal chromosomes in the gamete example, what is the probability of
a gamete having at least one maternal chromosome?
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Answer:
P(at least one maternal chromosome) = 1 - P(no maternal chromosomes) = 1 - 1/2^23
6. For a type of tissue the probability of a randomly selected cell being in a particular stage
of the cell cycle is given by the following table which describes the sample space:
7. Many types of color blindness are what are called X linked, that is determined by genes on
the X chromosome. Suppose a woman is carrying one X chromosome with the gene for a
particular type of color blindness; her other X chromosome does not have this gene. If she is
married to a man who does not have this gene on his X chromosome, consider the following:
A. What is the probability that her first child will carry the X chromosome with the gene
associated with color blindness?
Hint: remember the male has sex chromosomes X,y. For now let 'X*' be an X chromosome
carrying the allele for color blindness and 'X' be the chromosome carrying the allele for
normal vision.
B. An individual will have color vision if he or she has at least on X chromosome without the
gene for color blindness. What is the probability that the child is color blind?
Hints: P(color blind) = 1 - P(at least one X chromosome without the gene for color blindness).
1 - P(at least one x chromosome without the gene for color blindness AND child is
female)*P(child is female) +
P(at least one x chromosome without the gene for color blindness AND child is male)*P(child
is male)