Odds, Permutations and Combinations and How to Calculate the Probability of an Event
What Is Probability Theory?
Probability theory is an interesting area of statistics concerned with the odds or chances of an event happening in a trial, e.g. getting a six when a dice is thrown or drawing an ace of hearts from a pack of cards. To work out odds, we also need to have an understanding of permutations and combinations. The math isn't terribly complicated, so read on and you might be enlightened!
Before we get started let's review a few key terms.
- Probability is a measure of the likelihood of an event occurring.
- A trial is an experiment or test. E.g., throwing a dice or a coin.
- The outcome is the result of a trial. E.g., the number when a dice is thrown, or the card pulled from a shuffled pack.
- An event is an outcome of interest. E.g., getting a 6 in a dice throw or drawing an ace.
What Is the Probability of an Event?
There are two types of probability, empirical and classical.
If A is the event of interest, then the probability of A occurring is denoted by P(A).
This is determined by carrying out a series of trials. So, for instance, a batch of products is tested and the number of faulty items is noted plus the number of acceptable items.
If there are n trials
and A is the event of interest
Then if event A occurs x times
P(A) = x / n
Example: A sample of 200 products is tested and 4 faulty items are found. What is the probability of a product being faulty?
So x = 4 and n = 200
Therefore P(faulty item) = 4 / 200 = 0.02
This is a theoretical probability which can be worked out mathematically.
If A is the event, then
P(A) = Number of ways the event can occur / The total number of possible outcomes
Example 1: What are the chances of getting a 6 when a dice is thrown?
In this example, there is only 1 way a 6 can occur and there are 6 possible outcomes, i.e. 1, 2, 3, 4, 5 or 6.
So P(6) = 1/6
Example 2: What is the probability of drawing a 4 from a pack of cards in one trial?
There are 4 ways a 4 can occur, i.e. 4 of hearts, 4 of spades, 4 of diamonds or 4 of clubs.
Since there are 52 cards, there are 52 possible outcomes in 1 trial.
So P(4) = 4 / 52 = 1 / 13
What Is the Expectation of an Event?
Once a probability has been worked out, it's possible to get an estimate of how many events will likely happen in future trials. This is known as the expectation and is denoted by E.
If the event is A and the probability of A occurring is P(A), then for N trials, the expectation is:
E = P(A) N
For the simple example of a dice throw, the probability of getting a six is 1/6.
So in 60 trials, the expectation or number of expected 6's is:
E = 1/6 x 60 = 10
Remember, the expectation is not what will actually happen, but what is likely to happen. In 2 throws of a dice, the expectation of getting a 6 (not two sixes) is:
E = 1/6 x 2 = 1/3
However, as we all know, it's quite possible to get 2 sixes in a row, even though the probability is only 1 in 36 (see how this is worked out later). As N becomes larger, the actual number of events which happen will get closer to the expectation. So for example when flipping a coin, if the coin isn't biased, the number of heads will be closely equal to the number of tails.
Success or Failure?
The probability of an event can range from 0 to 1.
P(Event) = Number of ways the event can occur / The total number of possible outcomes
So for a dice throw
P(getting a number between 1 and 6) = 6 / 6 = 1 (since there are 6 ways you can get "a" number and 6 possible outcomes)
P(getting a 7) = 0 / 6 = 0 (there are no ways the event 7 can occur in any of the 6 possible outcomes)
P(getting a 5) = 1 / 6 (only 1 way of getting a 5)
If there are 999 failures in 100 samples
Empirical probability of failure = P(failure) = 999/1000 = 0.999
A probability of 0 means that an event will never happen.
A probability of 1 means that an event will definitely happen.
In a trial, if event A is a success, then failure is not A (not a success)
And P(A) + P(not A) = 1
Addition Law of Probability
For mutually exclusive (they can't occur simultaneously) events A and B
P(A or B) = P(A) + P(B)
Example: A dice is thrown and a card is drawn from a pack, what is the possibility of getting a 6 or an ace?
P(getting a six) is 1/6
There are 52 cards in a pack and four ways of getting an ace.
P(getting an ace) is 4/52 = 1/13
So P(getting a six or getting an ace) = P(getting a six) + P(getting an ace) = 1/6 + 1/13
Remember in these type of problems, how the question is phrased is important. So the question was to determine the probability of one event occurring "or" the other event occurring and so the addition law of probability is used.
Multiplication Law of Probability
For independent (the first trial doesn't affect the second trial) events A and B
P(A and B) = P(A) x P(B)
Example: A dice is thrown and a card drawn from a pack, what is the probability of getting a 5 and a spade card?
P(getting a 5) = 1/6
There are 52 cards in the pack and 4 suits or groups of cards, aces, spades, clubs and diamonds. Each suit has 13 cards, so there are 13 ways of getting a spade.
So P(drawing a spade) = 13/52 = 1/4
P(getting a 5 and drawing a spade) = P(getting a 5) x P(drawing a spade) = 1/6 x 1/4 = 1/24
Again it's important to note that the word "and" was used in the question, so the multiplication law was used.
Independent and Dependent Events
Events are independent when the occurrence of one event doesn't affect the probability of the other event.
So if a card is drawn from a pack, the probability of an ace is 4/52 = 1/13.
If the card is replaced, the probability of drawing an ace is still 1/13.
Events are dependent if the occurrence of the first event affects the probability of occurrence of the second event.
If an ace is drawn from a pack and not replaced, there are only 3 aces left and 51 cards remaining, so the probability of drawing a second ace is 3/51.
For two events A and B where B depends on A, the probability of Event B occurring after A is denoted by P(B|A).
Permutations and Combinations
To solve more difficult problems and derive an expression for the probability of a general binomial distribution, we need to understand the concept of permutations and combinations. I won't go into the mathematics of the derivation, but basically the expression is derived from the equation for working out combinations.
A Permutation Is an Arrangement
A permutation is a way of arranging a number of objects. So, for instance, if you have the letters A, B, and C then all the possible permutations are:
ABC, ACB, BAC, BCA, CAB, CBA
Note that BA is a different permutation to AB.
If you have n objects, there are n factorial number of ways of arranging them, written as n!
n! = n x (n-1) x (n-2) .... x 3 x 2 x 1
The reason for this is because for the first position, there are n choices, and for each of these choices, there are (n-1) choices for the second place (because 1 choice was used up for the first place), and for each of the choices in the first two places, (n-3) choices for the third place and so on.
In the example above, the 3 letters A, B, C could be arranged in 3! = 3 x 2 x 1 = 6 ways
In general, if n objects are selected r at a time then, the number of permutations is:
n! / (n-r)!
This is written as nPr
Example: 2 letters are chosen from the set of letters A, B, C, D. How many ways can the 2 letters be arranged?
There are 4 letters so n =4 and r = 2
nPr = 4P2 = 4! / (4 - 2)! = 4! / 2! = 4 x 3 x 2 x 1 / 2 x 1 = 12
A Combination Is a Selection
A combination is a way of selecting objects from a set without regard to the order of the objects. So again if we have the letters A, B and C and select 3 letters from this set, there is only 1 way of doing this, i.e. select ABC.
If we select 2 letters at a time from ABC, all the possible selections are:
AB, AC, and BC
Remember, BA is the same selection as AB etc.
In general, if you have n objects in a set and make selections r at a time, the total possible number of selections is:
nCr = n! / ((n - r)! r!)
Example: 2 letters are chosen from the set ABCD. How many combinations are possible?
There are 4 letters ao n = 4 and r = 2
nCr = 4C2 = 4! / ( (4 - 2)! x 2!) = 4! / (2! x 2!)
= 4 x 3 x 2 x 1 / ( (2 x 1) x (2 x 1) ) = 6
General Binomial Distribution
In a trial, an event could be getting heads in a coin throw or a six in a throw of a dice.
If the occurrence of an event is defined as a success, then
Let the probability of success be denoted by p
Let the probability of non-occurrence of the event or failure be denoted by q
p + q = 1
Let the number of successes be r
And n is the number of trials
Example: What are the chances of getting 3 sixes in 10 throws of a dice?
There are 10 trials and 3 events of interest, i.e. successes so:
n = 10
r = 3
The probability of getting a 6 in a dice throw is 1/6, so:
p = 1/6
The probability of not getting a dice throw is:
q = 1 - p = 5/6
P(3 successes) = 10! / ((10 - 3)! 3!) x (5/6)(10 - 3) x (1/6)3
= 10! / (7! x 3!) x (5/6)7 x (1/6)3
= 3628800 / (5040 x 6) x (78125 / 279936) x (1/216)
Winning the Lottery! How to Work out the Odds
We would all like to win the lottery, but the chances of winning are only slightly greater than 0. However "If you're not in, you can't win" and a slim chance is better than none at all!
Take, for example, the California State Lottery. A player must choose 5 numbers between 1 and 69 and 1 Powerball number between 1 and 26. So that is effectively a 5 number selection from 69 numbers and a 1 number selection from 1 to 26. To calculate the odds, we need to work out the number of combinations, not permutations, since it doesn't matter what way the numbers are arranged to win.
The number of combinations of r objects is nCr = n! / ((n - r)! r!)
n = 69
r = 5
nCr = 69C5 = 69! / ( (69 - 5)! 5!) = 69! / (64! 5!) = 11,238,513
So there are 11,238,513 possible ways of picking 5 numbers from a choice of 69 numbers.
Only 1 Powerball number is picked from 26 choices, so there are only 26 ways of doing this.
For every possible combination of 5 numbers from the 69, there are 26 possible Powerball numbers, so to get the total number of combinations, we multiply the two combinations.
So the total possible number of combinations = 11,238,513 x 26 = 292,201,338 or roughly 293 million and the probability of winning is 1 in 293 million.
Engineering Mathematics by K.A. Stroud - An Excellent Textbook!
Engineering Mathematics by K.A. Stroud is an excellent math textbook for both engineering students and anyone with a general interest in mathematics. The material has been written for part 1 of BSc. Engineering Degrees and Higher National Diploma courses.
A wide range of topics are covered including matrices, vectors, complex numbers, calculus, calculus applications, differential equations and series. The text is written in the style of a personal tutor, guiding the reader through the content, posing questions and encouraging them to provide the answer. Personally, I've found it really easy to follow.
It also covers a more in-depth treatment of probability theory as outlined in this article plus a section on statistics.
This book basically makes learning mathematics fun!
Note: Second hand 1987 editions of this text book are available on Amazon for only about $6
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© 2016 Eugene Brennan