Download Introduction to Probability - Buisness Management - Lecture Notes and more Study notes Business Administration in PDF only on Docsity!
Chapter 5
Introduction to Probability
5.1 Introduction
Probability is the language we use to model uncertainty. We all intuitively understand that few things in life are certain. There is usually an element of uncertainty or randomness around out- comes of our choices. In business this uncertainty can make all the difference between a good investment and a poor one. Hence an understanding of probability and how we might incorporate this into our decision making processes is important. In this chapter, we look at the logical basis for how we might express a probability and some basic rules that probabilities should follow. In the next chapter, we look at how we can use probabilities to aid decision making.
5.1.1 Definitions
We often use the letter P to represent a probability. For example, P (Rain) would be the probability of the event of it raining.
Experiment An experiment is an activity where we do not know for certain what will happen but we will observe what happens. For example:
- We will ask someone whether or not they have used our product.
- We will observe the temperature at mid day tomorrow.
- We will toss a coin and observe whether it shows “heads” or “tails”.
Outcome An outcome, or elementary event , is one of the possible things that can happen. For example, suppose that we are interested in the (UK) shoe size of the next customer to come into a shoe shop. Possible outcomes include “eight”, “twelve”, “nine and a half” and so on. In any experiment, one and only one outcome occurs.
Sample space The sample space is the set of all possible outcomes. For example it could be the set of all shoe sizes.
Event An even is a set of outcomes. For example “the shoe size of the next customer is less than 9” is an event. It is made of of all of the outcomes where the shoe size is less than 9. Of course an event might contain just one outcome.
Probabilities are usually expressed in terms of fractions or decimal numbers or percentages. There- fore we could express the probability of it raining today as
P (Rain) =
All probabilities are measured on a scale ranging from zero to one. The probabilities of most events lie strictly between zero and one as an event with probability zero is an impossible event and one with probability one is a certain event.
The collection of all possible outcomes, that is the sample sapce, has a probability of 1. For example, if an event consists of only two outcomes success or failure then the probability of either a success or a failure is 1. That is P (success or f ailure) = 1.
Two events are said to be mutually exclusive if both can not occur simultaneously. In the example above, the outcomes success and a failure are mutually exclusive.
Two events are said to be independent if the occurence of one does not affect the probability of the second occurring. For example, if you toss a coin and look out of the window, it would be reasonable to suppose that the events “get heads” and “it is raining” would be independent. How- ever, not all events are independent. For example, if you go into the Students’ Union Building and pick a student at random, then the events “the student is female” and “the student is studying en- gineering” are not independent since there is a greater proportion of male students on engineering courses than on other courses at the University (and this probably applies to those students found in the Union).
5.2 How do we measure Probability?
There are three main ways in which we can measure probability. All three obey the basic rules described above. Different people argue in favour of the different views of probability and some will argue that each kind has its uses depending on the circumstances.
5.2.1 Classical
If all possible outcomes are “equally likely” then we can adopt the classical approach to measuring probability. For example if we tossed a fair coin, there are only two possible outcomes, a head or a tail both of which are equally likely and hence
P (Head) =
and P (T ail) =
The underlying idea behind this view of probability is symmetry. In this example, there is no reason to think that the outcome Head and the outcome Tail have different probabilities and so
undertake the journey. Similarly, the odds given by bookmakers on a horse race reflect people’s beliefs about which horse will win. This probability does not fit within the frequentist definition as the race cannot be run a large number of times.
One potential difficulty with using subjective probabilities is that it is subjective. So the probabil- ities which two people assign to the same event can be different. This becomes important if these probabilities are to be used in decision making. For example, if you were deciding whether to launch a new product and two people had very different ideas about how likely success or failure of this product was, then the decision to go ahead could be controversial. If both individuals as- sessed the probability of success to be 0.8 then the decision to go ahead could easily be based on this belief. However, if one said 0.8 and the other 0.3, then the decision is not straightforward. We would need a way to reconcile these different positions.
Subjective probability is still subject to the same rules as the other forms of probability, namely that all probabilities should be positive and that the probability of all outcomes should sum to one. Therefore, if you assess P (Success) = 0. 8 then you should also assess P (F ailure) = 0. 2.
5.3 Laws of Probability
5.3.1 Multiplication Law
The probability of two independent events E 1 and E 2 both occurring can be written as
P (E 1 and E 2 ) = P (E 1 ) × P (E 2 ).
For example, if the probability of throwing a six followed by another six on two rolls of a die is calculated as follows. The outcomes of the two rolls of the die are independent. Let E 1 denote a six on the first roll and E 2 a six on the second roll. Then
P (two sixes) = P (E 1 and E 2 ) = P (E 1 ) × P (E 2 ) =
×
This method of calculating probabilities extends to when there are many independent events
P (E 1 and E 2 and · · · and En) = P (E 1 ) × P (E 2 ) × · · · × P (En).
(There is a more complicated rule for multiplying probabilities when the events are not indepen- dent).
5.3.2 Addition Law
The multiplication law is concerned with the probability of two or more independent events oc- curring. The addition law describes the probability of any of two or more events occurring. The addition law for two events E 1 and E 2 is
P (E 1 or E 2 ) = P (E 1 ) + P (E 2 ) − P (E 1 and E 2 ).
This describes the probability of either event E 1 or event E 2 happening.
Consider the following information: 50 percent of families in a certain city subscribe to the morn- ing newspaper, 65 percent subscribe to the afternoon newspaper, and 30 percent of the families subscribe to both newspapers. What proportion of families subscribe to at least one newspaper?
We are told P (Morning) = 0. 5 , P (Afternoon) = 0. 65 and P (Morning and Afternoon) = 0. 3. Therefore
P (at least one paper) = P (Morning or Afternoon) = P (Morning) + P (Afternoon) − P (Morning and Afternoon) = 0.5 + 0. 65 − 0. 3 = 0. 85.
So 85% of of the city subscribe to at least one of the newspapers.
A more basic version of the rule works where events are mutually exclusive: if events E 1 and E 2 are mutually exclusive then P (E 1 or E 2 ) = P (E 1 ) + P (E 2 ).
This simplification occurs because when two events are mutually exclusive they cannot happen together and so P (E 1 and E 2 ) = 0.
These two laws are the basis of more complicated problem solving we will see later.
5.3.3 Example
A building has three rooms. Each room has two separate electric lights. There are thus six electric lights altogether. After a certain time there is a probability of 0.1 that a given light will have failed and all light are independent of all other lights. Find the probability that, after this time, there is at least one room in which both lights have failed.
Solution
For a given light, the probability that it has failed is 0.1.
For a given room, the probability that both lights have failed is
0. 1 × 0 .1 = 0. 01.
For a given room, the probability that it is not true that both lights have failed, that is the probability that at least one of the two lights is working, is
The probability that at least one light is working in every one of the three rooms (that is, in Room A and in Room B and in Room C) is
Student Height Weight Shoe Student Height Weight Shoe Number Sex (m) (kg) Size Number Sex (m) (kg) Size 1 M 1.91 70 11.0 10 M 1.78 76 8. 2 F 1.73 89 6.5 11 M 1.88 64 9. 3 M 1.73 73 7.0 12 M 1.88 83 9. 4 M 1.63 54 8.0 13 M 1.70 55 8. 5 F 1.73 58 6.5 14 M 1.76 57 8. 6 M 1.70 60 8.0 15 M 1.78 60 8. 7 M 1.82 76 10.0 16 F 1.52 45 3. 8 M 1.67 54 7.5 17 M 1.80 67 7. 9 F 1.55 47 4.0 18 M 1.92 83 12.
Find the probabilities for the following events.
(a) The student is female. (b) The student’s weight is greater than 70kg., (c) The student’s weight is greater than 70kg. and the student’s shoe-size is greater than 8, (d) The student’s weight is greater than 70kg. or the student’s shoe-size is greater than 8.
