• UNIT 16 : Elementary probability

     Key unit competence

    Use counting techniques and concepts of probability to determine the probability of possible outcomes  of events occurring under equally likely assumptions.

    Learning objectives 



    16.1 Concepts of probability

    Activity 16.1

    In groups, discuss the following. 

    Have you ever watched people gambling? 

    What do they rely on to win? What are the problems associated with gambling? Present your findings to the rest of the class.

    Many events cannot be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.



    Probability




    Sample space and events 

    The set S of all possible outcomes of a given experiment is called the sample space. A particular outcome, i.e., an element of S, is called a sample point. Any subset of the sample space is called an event. The event {a} consisting of a single element of S is called a simple event. 

    Experiment or trial: an action where the result is uncertain. 

    Tossing a coin, throwing dice, seeing what fruits people prefer are all examples of experiments. 

    Sample space: all the possible outcomes of an experiment 

    Choosing a card from a deck 

    There are 52 cards in a deck (not including Jokers) So the sample space is all 52 possible cards: {Ace of Hearts, 2 of Hearts, and so on... } 

    Event: a single result of an experiment 

    • Getting a Tail when tossing a coin is an event 

    • Rolling a “5” is an event. An event can include one or more possible outcomes: 

    • Choosing a “King” from a deck of cards (any of the 4 Kings) is an event • Rolling an “even number” (2, 4 or 6) is also an event 

    Anne wants to see how many times a “double” comes up when throwing 2 dice. Each time Anne throws the 2 dice is an experiment. It is an experiment because the result is uncertain.

    The event Anne is looking for is a “double”, where both dice have the same number. It is made up of these 6 sample points: 

    (1,1), (2,2), (3,3), (4,4), (5,5) and (6,6) or S ={(1,1), (2,2), ... .} 

    The sample space is all possible outcomes (36 sample points): 

    (1,1), (1,2), (1,3), (1,4) ... (6,3), (6,4), (6,5), (6,6) 


    Probability of an event under equal assumptions




    Complementary event 

    If E is an event, then E’ is the event which occurs when E does not occur. Event E and E’ are said to be complementary events.


    Consider two different events, A and B. which may occur when an experiment is performed. The event A ∪ B  is the event which occurs if A or B or both A and B occurs, i.e.. at least one of A or B occurs.

    The event A ∩ B  is the event which occurs when both A and B occurs. 

    The event A – B is the event which occurs when A occurs and B does not occur. 

    The event A′ is the event which occurs when A does not occur.




    Permutations and combinations in probability theory

    Activity 16.2 

    Discuss the following scenario in groups and present your findings to the rest of the class. 

    If there are 4 women and 3 men and you wanted them seated in a row, what is the probability that the men will be seated together?

    Permutations and combinations can be used to find probabilities of various events particularly when large sample sizes occur. In everything we do, we have to use the formula 






    16.2 Finite probability spaces 




    16.3 Sum and product laws

     


    Mutually exclusive events 

    Events A and B are said to be mutually exclusive if the events A and B are disjoint i.e. A and B cannot occur at the same time.

    For mutually exclusive events, A ∩ B = ∅. P(A ∩ B) = P(∅) = 0; and so the addition law reduces to P(A ∪ B) = P(A) + P(B).



    Independent events 

    Two events are independent if the occurrence or non occurrence of one of them does not affect the occurrence of the other. Otherwise, A and B are dependent. For independent events, the probability that they both occur is given by the Following product law: 

    P(A ∩ B) = P(A) × P(B)







    16.4 Conditional probability 

    The probability of an event B occuring given that A has occurred is called the conditional probability of B given A and is written P(B|A). 

    Here P(B|A) is the probability that B occurs considering A as the sample space, and since the subset of A in which B occurs is A ∩ B, then 


    The general statement of the multiplication law is obtained by rearranging this result: 

    P(A ∩ B) = P(A) × P(B|A) 

    Thus the probability that the two events will both occur is the product of the probability that one will occur and the conditional probability that the other will occur given that the first has occurred. As A and B are interchangeable, we can also write P(A ∩ B) = P(B) × P(A|B) 

    If A and B are independent, then the probability of B is not affected by the occurrence of A and so P(B|A)=P(B) giving P(A ∩ B) = P(A) × P(B) which is our definition of independence.


    A tree diagram can be quite useful in the calculation of certain probabilities. The following example illustrates the method.






















    Supplementary, interactive questions served by Siyavula Education.

    Note: Questions will open in a new window or tab.

    Concepts of probability

    1. Probability of arrangements of PINs
    2. Probability basics
    3. Mutually exclusive events
    4. Mutually exclusive sets
    5. Mutually exclusive events: phone numbers

    Supplementary, interactive questions served by Siyavula Education.

    Note: Questions will open in a new window or tab.

    Finite probability spaces

    1. Probability of picking a card
    2. Probability for rolling a die
    3. Probabilities with more complex outcomes
    4. Dice experiment
    5. Go fish!

    Supplementary, interactive questions served by Siyavula Education.

    Note: Questions will open in a new window or tab.

    Sum and product laws

    1. Using the probability identity
    2. Probability identities
    3. Prove that two events are mutually exclusive
    4. Mutually exclusive events
    5. Working with probabilities of events
    6. Dependent and independent events
    UNIT 15 : Combinatorics