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Permutation, Combination and Probability

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Chapter 6 : Permutation, Combination and Probability

Factorial arrow_upward

  • Factorial of n (written as n!) is the product of all positive integers less than or equal to n.
  • n! = (n)(n - 1)(n - 2). . . (3)(2)(1)


        3! = (3)(2)(1) = 6

        4! = (4)(3)(2)(1) = 24

    Permutation arrow_upward

  • A Permutation is an arrangement of the items of a group where order matters.
  • Example:
  • Consider the following:
    • Given 4 people, B, M, S and A, how many different ways can these four people be arranged where order matters?
    • We have the following:

























  • There are 24 ways to arrange the four people, four at a time, which is 4!.

  • Property:

  • There are n! ways to arrange n objects in groups of n at a time.

  • Permutation Formula arrow_upward

  • The number of permutations of n objects taken r at a time is:
  • Example:
  • Find the number of ways to arrange people in groups of  at a time where order matters.
  • Solution:

  • There are 24 ways to arrange 4 items taken 3 at a time when order matters.

  • Combinations arrow_upward

  • A Combination is one of the different arrangements of items of a group where order does not matter.
  • Example:
  • Suppose we want to take four people, B, M, S and A, and arrange them in groups of three at a time where order does not matter.
  • The following demonstrates all the possible arrangements:

  • BMS




  • There are 4 ways to arrange 4 people in groups of 3 at a time.

  • Combination Formula arrow_upward

  • The number of combinations of a group of n objects taken r at a time is:
  • Example:
  • Find the number of ways to take 4 people and place them in groups of 3 at a time where order does not matter.
  • Solution:
  • Use the combination formula
  • There are 4 ways to arrange 4 items taken 3 at a time when order does not matter.

  • Theorem arrow_upward

  • Suppose that a sequence S of n items has n1 identical objects of type 1, n2 identical objects of type 2 and ni identical objects of type i.
  • Then, the number of orderings of S is:
  • Example:
  • The word MISSISSIPPIhas the following number of orderings:
  • Solution:

  • The first ! in the denominator is due to the letter I repeated 4 times.
  • The second 4! in the denominator is due to the letterS repeated 4 times.
  • The 2! in the denominator is due to the letter P repeated 2 times.

    Random Experiment arrow_upward

  • An action that leads to one of several possible outcomes.

  • Experiment


    Flip a coin

    Heads, Tails

    Sample Space arrow_upward

  • In any number of given trials, the set of all possible outcomes is called the Sample Space.
  • The sample space is often represented by the letter S.
  • Example:
  • In tossing a coin, there are two possible outcomes.
    • Sample Space S = {heads, tails}
  • or S = {H, T}
  • Example:
  • The Sample Space for rolling a die once is:
    • Sample Space S = {1, 2, 3, 4, 5, 6}

    Events arrow_upward

  • A collection of outcomes for the experiment, that is, any subset of the sample space.
  • Examples:
  • Getting a Tail when tossing a coin is an event.
  • Rolling an “even number” (2, 4 or 6) is an event.

  • Probability for Equally Likely Outcomes arrow_upward

  • Suppose an experiment has N possible outcomes, all equally likely.
  • Then the probability that a specified event occurs will be equal to the number of ways that the event can occur to the total number of possible outcomes.

  • Probability Notation arrow_upward

  • If E is an event, then P(E) stands for the probability that an eventE occurs.
  • It is read as “the probability of E”.
  • Example:
  • What is the probability of getting the number 5 if a die (balanced) is rolled?
  • Solution:
  • A die has 6 numbers.
  • There is only one 5 on a die, so the probability of getting a number 5 is:
  • P(5) = 1/6

  • Probability Rules arrow_upward

  • The probability of any event occurring will always be a number between zero and one.
  • When an event cannot occur the probability will be 0.
  • When an event is certain to occur the probability is1.
  • The sum of all the probabilities of all the possible outcomes is 1.
  • The probability of an event happening added with the probability of the event not happening is always 1.

  • Mutually Exclusive Events arrow_upward

  • If the events A and Bare mutually exclusive, then both events cannot occur simultaneously.
  • A and B do not share any outcomes:
  • P(A and B) = 0
  • For Mutually Exclusive Events:

    Non Mutually Exclusive Events arrow_upward

  • Two or more events are said to be "Non Mutually Exclusive" if these events can occur simultaneously. That is the occurrence of one does not prevent the occurrence of the others in all cases.

  • Independent Events arrow_upward

  • If A and Bare independent events, then the chance of Aoccurring does not affect the chance of B occurring and vice versa.

  • Dependent Events arrow_upward

  • When events are dependent, each possible outcome is related to the other.
  • P(A and B) = P(A) × P(B given that A has happened)

  • Conditional Probability arrow_upward

  • Probability of an event occurring given that another event has already occurred is called as Conditional Probability.
  • The probability that event A occurs, given that event B has already occurred is

  • Union of Events arrow_upward

  • If either event A or event B or both occur on a single performance of an experiment, this is called the Union of the Events A and B, and is denoted by.
  • The union of A and B is the whole colored area.
  • If two events are mutually exclusive then the probability of either occurring is:

  • If the events are not mutually exclusive, then:

  • Intersection of Events arrow_upward

  • If both events happen simultaneously it is written as  and it is read asA intersection B.
  • The intersection of A and Bis the purple overlapping area.
  • If two events, A and B are independent, the joint probability is:

  • Complement of Events arrow_upward

  • The Complement of Events is the set of all outcomes of an experiment that are not included in an event.
  • The complement of event A is written as Ac .
  • If P(A)is the probability of happening an event then the probability of complementary event is:
  • Or

  • Bayes’ Theorem arrow_upward

  • Bayes' Theorem relates the conditional and marginal probabilities of events A and B, provided that the probability of Bdoes not equal zero.
  • Where,
    • P(B|A) is the conditional probability of B, given A.
    • P(A) is the prior probability.
    • P(B) is the prior or marginal probability of B.
    • P(A|B) is the conditional probability of A, given B.

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