Probability is a numerical value that measures the uncertainty that a particular even will occur. The probability of an event represents the portion of times under similar circumstances that the event can be expected to occur. This chapter will present the rules one uses to determine probability of events under varying circumstances.
1. Know what is meant by the term probability.
The uncertain outcome of an experiment is called an event.
There are two common outcomes associated with the tossing of a coin: the two events are: "Head" or "Tail". In probability studies, an event is the outcome of a random experiment. For example, when you toss a coin, you are not certain which event will occurs (randomness), but you can expect that you will either get a "Head" or a "Tail".
Probability is the proportion of times that the event can be expected to occur.
The probability of an event is often written as Prob[event] or Pr[event]
or P(event).
| Note:
the probability of an event, say getting a Tail when tossing a fair coin
is the number of ways or times a Tail can occur divided by the total number
of possible outcomes.
Probability of A = Number of times A divided by Total number of possible outcomes. When a coin is tossed the probability of Tails, Prob(Tail) or P(T) = ½ Since the total possible outcome is Head or Tails, we get 2 for possible outcomes. If H denotes Head and T denotes Tail, what is the probability of getting two Heads if you toss two coins? (it does not matter which coin get heads or tails ) Possible outcomes (sample space) = HH, HT, TH, TT, so the sample space = 4 and there is 1 way of getting two heads , so Prob(2 Heads) or P(HH) =0.25 What is the probability of getting even numbers , P(even), when you roll a dice? (Answer, ½ , since sample space is 1, 2, 3, 4, 5, 6 and even numbers from these are: 2, 4, 6) |
Over the years the study of probability have been associated with gambling but have since evolve into a branch of mathematics dealing with uncertainties of events; however, gambling experiment offers many usefully and simple ways to illustrate the concepts of probability.
2. Know what is meant by the term sample space.
The sample space is all the possible outcomes of an experiment.
The sample space is written mathematically with the use of { }. For example, the possible outcomes or sample space for tossing a fair coin can be written as:
Sample space = {Head, Tail}
The sample space for student major at Andrews University is:
Sample space = {Accounting, Mathematics, Education, Leadership, English, etc.}
The sample space for drawing a card for a random desk of card can be written as:
Sample space = {K
,
Q
,
..., 3
,
2
,
A
,
K
,.....,A
}
The entire sample space for selecting randomly from a deck
of 52 ordinary playing cards are:
| Denomination | Spades
(Black) |
Hearts
(red) |
Clubs
(black) |
Diamonds
(red) |
| King | K |
K |
K |
K |
| Queen | Q |
Q |
Q |
Q |
| Jack | J |
J |
J |
J |
| 10 | 10 |
10 |
10 |
10 |
| 9 | 9 |
9 |
9 |
9 |
| 8 | 8 |
8 |
8 |
8 |
| 7 | 7 |
7 |
7 |
7 |
| 6 | 6 |
6 |
6 |
6 |
| 5 | 5 |
5 |
5 |
5 |
| 4 | 4 |
4 |
4 |
4 |
| 3 | 3 |
3 |
3 |
3 |
| Deuce (2) | 2 |
2 |
2 |
2 |
| Ace | A |
A |
A |
A |
The face cards are the K, Q, and J (Queen, King and Jack, example
Q
is called Queen of Clubs).
3. Know how to calculate the probability of a simple event.
The probability of an event is often written as Prob[event] or Pr[event] or P(event).
The Pr[event] is a fraction or decimal between 0 and 1.
The probability of equally likely events (random) is the proportion of times the event can occurs or:
What is the Pr[K] (probability of selecting a king from a deck of cards)?
Since the Number of ways a king can be selected from a deck of cards are 4 ways:
{K
,
K
,
K
,
K
}
and the sample space is 52 (the sample space} then:
,
So Pr[K] = 0.0769 or 7.79% (see decimal to percent
conversion).
Let's illustrate the concept of elementary events by an example.
Example: If three coins were tossed at the same time, a penny, a nickel and a dime and since all three coins will land about the same time. The outcome of each coin is a random event and is called an elementary event.
So elementary events are independent or separate events.
A composite event is an event made up of a grouping of elementary events, the possible outcomes of all three coins tossed together is a composite event.
What if one was to ask for the probability of the dime being Head?
Let us study the sample space from an experiment of tossing 3 coins;
if H represents Head, T = Tails and subscripts p, n and d represents, penny,
nickel and dime respectively, then the sample space is:
| Event, i | Penny | Nickel | Dime |
| 1 | H | H | H |
| 2 | H | H | T |
| 3 | H | T | H |
| 4 | H | T | T |
| 5 | T | H | H |
| 6 | T | H | T |
| 7 | T | T | H |
| 8 | T | T | T |
These are the possible outcomes of tossing three coins and is found by iteration, writing the possibilities without trying to repeat that already written and only writing that which is possible.
The possible ways that a Dime can be a Head from the experiment above is 4 or
{HpHnHd , HpTnHd , TpHnHd , TpTnHd } Here each order represents the order of the coins - penny, nickel and dime.
So Pr[dime is a Head] = 4/8 = ½ (toss of three coins experiment)
Problem: What would be the possible outcomes or sample space
of tossing two coins?
| Note: In most of the research that you will conduct in your academic study, the probability of an event will be related to proportions or similar to the relative frequencies of groups of data or the proportion of times the event occurred. |
What is the probability of students selecting soup for lunch if a study
of 604 students during lunch time for 2 months shows the following results?
| Lunch Selection | Number of Students |
| Entrée (No Soup) | 420 |
| Soup Only | 54 |
| Soup plus something else | 130 |
Total number selecting soup = 54 + 130 = 184
Sample space = 420+54+130= 604
Pr[soup] = 184/604=0.3046 or 30.46%
How would you use the information to plan for 1000 students eating at the cafeteria next semester?
(hint use 1000 as the sample space and 30.46% as the Probability).
The event set is the possible outcome of an event. It can have the following values:
(1) A single element: Queen of Hearts from a deck of cards:
{Q
}
(2) Multiple element also called a Finite set: Dimes is a Head in 3 coin toss: {HpHnHd , HpTnHd , TpHnHd , TpTnHd }
(3) An impossible event: Not getting a 1, 2, 3, 4, 5,
6 in the roll of a die: { } or 
,
this is called the empty set of null set.
(4) Impossible event: are also denoted by 0, Pr[impossible event] = 0
(5) Certain event: An even that is always; the sun rising tomorrow, Pr[certain event] = 1
Pr[certain event] = 1
4. Know the Law of Large Numbers.
The Law of Large Numbers states that the probability of an event deviating significantly from its expected or theoretical probability becomes smaller as the number of repetitions of the experiment increases.
That is, the larger your experimental trials or sample size the more certain your results will reflect the expected or theoretical probabilities.
Example: It is possible to toss a fair coin and get 10 Heads in a row; however, if you continue to toss the coin say 200 times, the probability of getting a Head will be closed to that expected, ½ .
5. Know the Fundamental Principles of Counting.
Finding the sample space, the possible outcomes of an experiment is
conveniently done by applying the Fundamental Principle of Counting.
The Fundamental
Principle of Counting: If the possible outcomes of an event A is nA
outcomes and event B is nB outcomes, then both events A and
B performed together have possible outcomes of 
. |
Example: The sample space for the toss of three coins, penny, nickel and dime is the product of the size of the sample space for each and is 2 x 2 x 2 = 8 = 23
Number of possibilities = kn, where k = number of equaly likely outcomes for each events and n = number of trials or sample.
Example: The sample space of tossing a dime (two possible outcomes) and a die (6 possible outcomes) is 2 x 6 = 12
Example: A quality inspector wishes to sample 10 items and to label after inspection each item into one of 3 ways: satisfactory, reworkable, or scrap. The number of possible sample outcomes is:
3x3x3x3x3x3x3x3x3x3 = 310 = 59049
Example: Suppose an inspector wants to check the accuracy of a process with either of the following outcomes: A meets specification or B does not meet specification. What is the Pr[ABABBABBBA] if 10 samples are taken?
Since each trial has same possible outcome: either A or B, k = 2
Sample space for 10 trials = 210 = 1,024
So Pr[ABABBABBBA]= 1 / 1024 = 0.0009765625
Workshop Problem: Probability of Events
If two fair dice is rolled one after the other. (a) What is the sample space? (b) What is the probability of getting a 2 on the first die? (c) What is the probability of getting a total of 7 from both dice?
Problem #2. The following is the result of an experiment showing
the number of panelists in each category of a consumer test group.
| Occupation | Family Income | Total | ||
| Low | Medium | High | ||
| Homemaker | 8 | 26 | 6 | 40 |
| Blue-collar Worker | 16 | 40 | 14 | 70 |
| White-collar Worker | 6 | 62 | 12 | 80 |
| Professional | 0 | 2 | 8 | 10 |
| Total | 30 | 130 | 40 | 200 |
If one person is selected at random from this group.
(a) Find the probability that the selected person is
(1) a homemaker (2) a white-collar worker
(3) a blue-collar worker (4) a professional
(b) Find the probability that the selected person's family income is (1) low; (2) medium or (3) high.
(c) Find the probability that the selected person is
(1) a white-collar worker with a high income
(2) a homemaker with a low income
(3) a professional with a medium income
6. Know the difference between objective and subjective probabilities.
An objective probability is a proportion or quantity obtained from repeatable random experiments, such as those discussed above.
A subjective probability is based on judgment and not obtained from non repeatable circumstances, for example, the probability of you getting an A in all the courses you are taking this semester or the probability it will rain tomorrow in you town. Some of you live in climate where there is no snow, but can you say for certain that it will never snow anywhere in your state for the next 100 years?