Probability Problems, Shortcut Tricks and Examples

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Probability In an experiment if ‘ n ‘ is the number of exhaustive cases and ‘m’ is the number of favourable cases of an event A. Then the probability of event A is denoted by P(A).

P(A) = \frac{Number\; of\; Favourable\; Cases}{Number \; of \; Exhaustive\; Cases\; }\: =\:\;\frac{m}{n}\: =\,: \frac{n(A)}{n(S)}

Probability Theory

  • Random Experiment : An experiment in which all possible out comes are known and the exact output cannot be predicated in advance is called Random Experiment.
    (i) Tossing a fair coin
    (ii) Rolling an unbiased dice
    (iii) Drawing a card from a pack of well shuffled cards
  • Trail : Conducting a Random Experiment once is known as a Trail
  • Outcome : The result of a Trail in the random experiment is called as Outcome
    EXAMPLE : In tossing a single coin outcomes are H and T 
  • Sample Space : A set of all possible outcomes of a random experiments is known as Sample Space
    ( i ) In the experiment of tossing a coin the sample space S H,T
    ( ii ) If Two coins are tossed then S HH,HT,TH,TT
    ( iii ) In throwing a die S 1,2,3,4,5,6 
  • Event : Any non-empty subset of a Sample Space is called an Event
    ( i ). in Tossing a single coin getting Head or Tail is an Event
    ( ii )Getting an Ace (or) Diamond from a pack of 52 cards is an event
  • Exhaustive Events : The total number of possible outcomes of an experiment is known as Exhaustive Events
    EXAMPLE : In the experiment of throwing a die the total number of possible outcomes = 6
  • Favourable Events : The number of event which favour the happing of the events are known as Favourable Cases (or) Events
    EXAMPLE : In tossing two dice the number of cases favourable to getting the sum 3 is (2, 1), (1, 2) i.e., 2
  • Mutually Exclusive Events : If two events have no common outcomes then they are called Mutually Exclusive
    EXAMPLE: In tossing a coin the events Head & Tail are mutually exclusive because Head & Tail cannot happen at the same time.
  • Independent Events : Two events are said to be independent if the happening of one event does not affect the happening of the other.
  • Dependent Events : Two events are said to be dependent if the happening of an event will affect the happening of the other event .
    EXAMPLE: If we draw a card from a pack of 52 cards and replace it before we draw a second card. The second card is independent of first one. If we don’t replace the first card before the second draw. The second draw depends on the first one.
  • Axioms of Probability : Let S be the sample space and A be the event i.e., A\: \sqsubseteq \: S
    1. 0\: \leq \: P(A)\: \leq \: 1\: ,\forall \: A\: \sqsubseteq \: S
    2. P(S) = 1
    3. If A and B are mutually exclusive ( Disjoint i.e., A\: \cap \: B\: =\: \Phi )
    P(A\cup B)= P(A)+P(B)
  • Results of Probability :
    . for any two events A & B , P(A\cup B)\ =P(A)+P(B)-P(A\cap B)
    2. If \bar{A} denotes compliment of event A then , P\bar{A} = 1-P(A)
    3. For any two events A & B , P(\bar{A}\cap B)=P(B)-P(A\cap B)
    4. For any three events A, B & C , (A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(B\cap C)-P(A\cap C)+P(A\cap B\cap C)


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