Probability concepts by Sibanand Pattnaik - Part 1


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    Random Experiment

    In our day to day life, we perform many activities which have a fixed result no matter any number of times they are repeated. For example given any triangle, without knowing the three angles, we can definitely say that the sum of measure of angles is 180°.

    We also perform many experimental activities, where the result may not be same, when they are repeated under identical conditions. For example, when a coin is tossed it may turn up a head or a tail, but we are not sure which one of these results will actually be obtained. Such experiments are called random experiments.

    An experiment is called random experiment if it satisfies the following two conditions:
    (i) It has more than one possible outcome.
    (ii) It is not possible to predict the outcome in advance.

    Outcomes & Sample Space

    A Possible result of a random experiment is called its outcome.

    Consider the experiment of rolling a die. The outcomes of this experiment are 1, 2, 3, 4, 5, or 6, if we are interested in the number of dots on the upper face of the die.

    The set of outcomes {1, 2, 3, 4, 5, 6} is called the sample space of the experiment.

    Thus, the set of all possible outcomes of a random experiment is called the sample space associated with the experiment. Sample space is denoted by the symbol S.

    Each element of the sample space is called a sample point. In other words, each outcome of the random experiment is also called sample point.

    Let's start with a very basic question.

    Q1) Two coins (a one rupee coin and a two rupee coin) are tossed once. Find a sample space.

    Clearly the coins are distinguishable in the sense that we can speak of the first coin and the second coin. Since either coin can turn up Head (H) or Tail(T), the possible outcomes may be
    Heads on both coins = (H,H) = HH
    Head on first coin and Tail on the other = (H,T) = HT
    Tail on first coin and Head on the other = (T,H) = TH
    Tail on both coins = (T,T) = TT
    Thus, the sample space is S = {HH, HT, TH, TT}

    ORDERED N UN-ORDERED PAIRS

    ORDERED PAIR : In case of ordered pairs HT,TH are different..
    UN - ORDERED PAIRS : In case of un ordered HT n TH will be considered as 1 pair...

    Basically we find unordered when the variables in the question are implicit.....In algebra "Ways" basically refers to finding UNORDERED n "Solutions" refers to finding ORDERED.

    Because in " ways" (1,100) n (100,1) is same but in case of "Solution" (1,100) n (100,1) are different solutions....We ll go through the details when we explore algebra part....in questions like find the number of integral solutions to the equation 1/a + 1/b = 1/k.....we ll see this in algebra....i just thought that you guys should know the concept of "ordered n unordered" so told it here

    Q2) Find the sample space associated with the experiment of rolling a pair of dice (one is blue and the other red) once. Also, find the number of elements of this sample space.

    Suppose 1 appears on blue die and 2 on the red die. We denote this outcome by an ordered pair (1,2). Similarly, if ‘3’ appears on blue die and ‘5’ on red, the outcome is denoted by the ordered pair (3,5).

    In general each outcome can be denoted by the ordered pair (x, y), where x is the number appeared on the blue die and y is the number appeared on the red die.

    Therefore, this sample space is given by
    S = {(x, y): x is the number on the blue die and y is the number on the red die}.
    The number of elements of this sample space is 6 × 6 = 36 and the sample space is given below:
    {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6)
    (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6)
    (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

    Q3) In each of the following experiments specify appropriate sample space
    (i) A boy has a 1 rupee coin, a 2 rupee coin and a 5 rupee coin in his pocket. He takes out two coins out of his pocket, one after the other.
    (ii) A person is noting down the number of accidents along a busy highway during a year.

    (i) Let Q denote a 1 rupee coin, H denotes a 2 rupee coin and R denotes a 5 rupee coin. The first coin he takes out of his pocket may be any one of the three coins Q, H or R. Corresponding to Q, the second draw may be H or R. So the result of two draws may be QH or QR. Similarly, corresponding to H, the second draw may be Q or R.
    Therefore, the outcomes may be HQ or HR. Lastly, corresponding to R, the second draw may be H or Q.
    So, the outcomes may be RH or RQ.
    Thus, the sample space is S={QH, QR, HQ, HR, RH, RQ}
    (ii) The number of accidents along a busy highway during the year of observation can be either 0 (for no accident ) or 1 or 2, or some other positive integer.
    Thus, a sample space associated with this experiment is S= {0,1,2,...}

    Q4) A coin is tossed. If it shows head, we draw a ball from a bag consisting of 3 blue and 4 white balls; if it shows tail we throw a die. Describe the sample space of this experiment.

    Let us denote blue balls by B1, B2, B3 and the white balls by W1, W2, W3, W4. Then a sample space of the experiment is
    S = { HB1, HB2, HB3, HW1, HW2, HW3, HW4, T1, T2, T3, T4, T5, T6}.
    Here HBi means head on the coin and ball Bi is drawn, HWi means head on the coin and ball Wi is drawn. Similarly, Ti means tail on the coin.

    NOTE:

    Remember one thing --- in probability it doesn't matter whether the balls are identical or distinct
    in p&c : if I ask you in how many ways you can pick 1 ball out of 3 identical balls -- there is exactly 1 way. but in probability. there are 3c1 = 3 ways
    in probability --- we always look for no. of distinct ways

    Q5) Consider the experiment in which a coin is tossed repeatedly until a head comes up. Describe the sample space.

    In the experiment head may come up on the first toss, or the 2nd toss, or the 3rd toss and so on till head is obtained. Hence, the desired sample space is S= {H, TH, TTH, TTTH, TTTTH,...}

    Event

    The appearance of a particular outcome, which we may find in the sample space, is an event. Any subset of the sample space is called an event.

    Occurrence of an event: Consider the experiment of throwing a die. Let E denotes the event “ a number less than 4 appears”. If actually ‘1’ had appeared on the die then we say that event E has occurred. As a matter of fact if outcomes are 2 or 3, we say that event E has occurred. Thus, the event E of a sample space S is said to have occurred if the outcome ω of the experiment is such that ω ∈ E. If the outcome ω is such that ω ∉ E, we say that the event E has not occurred.

    Types of events:Events can be classified into various types on the basis of the elements they have.

    Impossible and Sure Events

    The empty set φ and the sample space S describe events. In fact φ is called an impossible event and S, i.e., the whole sample space is called the sure event. To understand these let us consider the experiment of rolling a die. The associated sample space is S = {1, 2, 3, 4, 5, 6}
    Let E be the event “ the number appears on the die is a multiple of 7”. Can you write the subset associated with the event E?
    Clearly no outcome satisfies the condition given in the event, i.e., no element of the sample space ensures the occurrence of the event E. Thus, we say that the empty set only correspond to the event E. In other words we can say that it is impossible to have a multiple of 7 on the upper face of the die. Thus, the event E = φ is an impossible event.
    Now let us take up another event F “the number turns up is odd or even”. Clearly F = {1, 2, 3, 4, 5, 6,} = S, i.e., all outcomes of the experiment ensure the occurrence of the event F. Thus, the event F = S is a sure event.

    Simple Event

    If an event E has only one sample point of a sample space, it is called a simple (or elementary) event.
    In a sample space containing n distinct elements, there are exactly n simple events.
    For example in the experiment of tossing two coins, a sample space is S={HH, HT, TH, TT}
    There are four simple events corresponding to this sample space. These areE1= {HH}, E2={HT}, E3= { TH} and E4={TT}.

    Compound Event

    If an event has more than one sample point, it is called a Compound event.
    For example, in the experiment of “tossing a coin thrice” the events
    E: ‘Exactly one head appeared’
    F: ‘Atleast one head appeared’
    G: ‘Atmost one head appeared’ etc.are all compound events.
    The subsets of S associated with these events are
    E={HTT,THT,TTH}
    F={HTT,THT, TTH, HHT, HTH, THH, HHH}
    G= {TTT, THT, HTT, TTH}

    Each of the above subsets contain more than one sample point, hence they are all compound events.

    Algebra of events

    Let A, B, C be events associated with an experiment whose sample space is S.

    Complementary Event

    For every event A, there corresponds another event A′ called the complementary event to A. It is also called the event ‘not A’.

    For example, take the experiment ‘of tossing three coins’. An associated sample space is
    S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
    Let A={HTH, HHT, THH} be the event ‘only one tail appears’

    Clearly for the outcome HTT, the event A has not occurred. But we may say that the event ‘not A’ has occurred. Thus, with every outcome which is not in A, we say that ‘not A’ occurs.

    Thus the complementary event ‘not A’ to the event A is
    A’= {HHH, HTT, THT, TTH, TTT}
    or A’= {ω: ω∈S and ω ∉A} = S – A.

    The Event ‘A or B’

    Recall that union of two sets A and B denoted by A ∪B contains all those elements which are either in A or in B or in both.

    When the sets A and B are two events associated with a sample space, then ‘A ∪B’ is the event ‘either A or B or both’. This event ‘A ∪B’ is also called ‘A or B’.
    Therefore Event ‘A or B’ = A ∪B = {ω: ω∈A or ω ∈B}

    The Event ‘A and B’

    We know that intersection of two sets A ∩B is the set of those elements which are common to both A and B. i.e., which belong to both ‘A and B’.

    If A and B are two events, then the set A ∩B denotes the event ‘A and B’.
    Thus, A ∩B = { ω: ω∈A and ω∈B}

    For example, in the experiment of ‘throwing a die twice’ Let A be the event ‘score on the first throw is six’ and B is the event ‘sum of two scores is atleast 11’ then A = {(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}, and B = {(5,6), (6,5), (6,6)} so A ∩B = {(6,5), (6,6)}

    Note that the set A ∩B = {(6,5), (6,6)} may represent the event ‘the score on the first throw is six and the sum of the scores is atleast 11’.

    The Event ‘A but not B’

    We know that A–B is the set of all those elements which are in A but not in B. Therefore, the set A–B may denote the event ‘A but not B’.We know that A – B = A ∩B´

    Q 6) Consider the experiment of rolling a die. Let A be the event ‘getting a prime number’, B be the event ‘getting an odd number’. Write the sets representing the events
    (i) Aor B
    (ii) A and B
    (iii) A but not B
    (iv) ‘not A’.

    Here S = {1, 2, 3, 4, 5, 6}, A = {2, 3, 5} and B = {1, 3, 5}
    Obviously
    (i) ‘A or B’ = A ∪B = {1, 2, 3, 5}
    (ii) ‘A and B’ = A ∩B = {3,5}
    (iii) ‘A but not B’ = A – B = {2}
    (iv) ‘not A’ = A′= {1,4,6}

    Mutually exclusive events

    In the experiment of rolling a die, a sample space is S = {1, 2, 3, 4, 5, 6}. Consider events, A ‘an odd number appears’ and B ‘an even number appears’

    Clearly the event A excludes the event B and vice versa. In other words, there is no outcome which ensures the occurrence of events A and B simultaneously. Here A = {1, 3, 5} and B = {2, 4, 6}
    Clearly A ∩B = φ i.e., A and B are disjoint sets.

    In general, two events A and B are called mutually exclusive events if the occurrence of any one of them excludes the occurrence of the other event, i.e., if they can not occur simultaneously. In this case the sets A and B are disjoint.

    Again in the experiment of rolling a die, consider the events A ‘an odd number appears’ and event B ‘a number less than 4 appears’

    Obviously A = {1, 3, 5} and B = {1, 2, 3}
    Now 3 ∈ A as well as 3 ∈ B
    Therefore, A and B are not mutually exclusive events.

    Exhaustive events Consider the experiment of throwing a die. We have S = {1, 2, 3, 4, 5, 6}. Let us define the following events

    A: ‘a number less than 4 appears’,
    B: ‘a number greater than 2 but less than 5 appears’
    and C: ‘a number greater than 4 appears’.
    Then A = {1, 2, 3}, B = {3,4} and C = {5, 6}. We observe that A ∪B ∪C = {1, 2, 3} ∪{3, 4} ∪{5, 6} = S.

    Such events A, B and C are called exhaustive events

    Exhaustive Events:

    If n events A1, A2, ........., An related to any particular sample space are such that if we take union of the sets of all the n events, sample space is formed. i.e.
    A1 υ A2 υ A3 υ ............... υ An = S
    If the union of all event is equal to sample space , the n it is known as exhaustive events

    Mutually Exclusive and Exhaustive Events.

    Events A1, A2, .......... An are said to be mutually exclusive and exhaustive if they satisfy the condition for mutual exclusion and exhaustiveness both.

    i.e. A1 υ A2 υ A3 ............ υ An = S

    and Ai ∩ Aj = Φ, where i = 1, 2, ........., n

    j = 1, 2, ........... n and i ≠ j.

    i.e. mutually exclusive and exhaustive events are those events of which union is equal to sample space and occurrence of any one of them excludes the possibility of occurrence of all others.

    Illustration:

    In rolling a die, events A1, A2, A3 and A4 are described as follows
    A1 : Number less then 2 occurs.
    A2 : 2 occurs
    A3 : odd number greater that 1 occur.
    A4 : even number greater than 2 occur.
    A1 : {1}
    A2 : {2}
    A3 : {3, 5}
    A4 : {4, 6}
    Clearly, A1 υ A2 υ A3 υ A4 = {1, 2, 3, 4, 5, 6} = S
    and A1 ∩ A2 = A1 ∩ A3 = A1 ∩ A4 = A2 ∩ A3 = A2 ∩ A4 = A3 ∩ A4 = Φ
    i.e. Ai ∩ Aj = Φ, i ≠ j, i, j = 1, 2, 3, ............ n.

    Q7) Two dice are thrown and the sum of the numbers which come up on the
    dice is noted. Let us consider the following events associated with this experiment
    A : ‘the sum is even’.
    B : ‘the sum is a multiple of 3’.
    C : ‘the sum is less than 4’.
    D : ‘the sum is greater than 11’.
    Which pairs of these events are mutually exclusive?

    There are 36 elements in the sample space S = {(x, y): x, y = 1, 2, 3, 4, 5, 6}.

    Q 8 ) A coin is tossed three times, consider the following events. A: ‘No head appears’, B: ‘Exactly one head appears’ and C: ‘Atleast two heads appear’.Do they form a set of mutually exclusive and exhaustive events?

    The sample space of the experiment is
    S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
    and A = {TTT}, B = {HTT, THT, TTH}, C = {HHT, HTH, THH, HHH}
    Now
    A ∪B ∪C = {TTT, HTT, THT, TTH, HHT, HTH, THH, HHH} = S
    Therefore, A, B and C are exhaustive events.
    Also, A ∩B = , A ∩C = and B ∩C =
    Therefore, the events are pair-wise disjoint, i.e., they are mutually exclusive.
    Hence, A, B and C form a set of mutually exclusive and exhaustive events.


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