Probability
Probability: A probability is a number that reflects the chance or likelihood that a particular event will occur. Probabilities can be expressed as proportions that range from 0 to 1, and they can also be expressed as percentages ranging from 0% to 100%. A probability of 0 indicates that there is no chance that a particular event will occur, whereas a probability of 1 indicates that an event is certain to occur. A probability of 0.45 (45%) indicates that there are 45 chances out of 100 of the event occurring.
Now, we will study basic terms viz random experiment, sample space, events etc. in probability with examples.
Experiment: An experiment is an operation which can produce well-defined outcomes. An experiment is an activity which has a fixed result no matter any number of times it is repeated. For example, “Given any triangle, without knowing the three angles, we can definitely say that the sum of the measure of angles is 180”. Now, we will discuss, what is a random experiment.
Random Experiment: An experiment whose outcomes can’t be predicted in advance is called a random experiment.
Sample Space: The set of all possible outcomes of an experiment is called a sample space. This is generally denoted by S.
Examples of Random Experiment:
Tossing a coin: When we toss a coin, outcome will be either head (H) or Tail (T). So, sample space is {H,T}
Throwing an unbiased die: When a die is thrown or rolled, the outcome is the number that appears on its upper face and it is a random integer from one to six, each value being equally likely. Sample space is {1,2,3,4,5,6}
Drawing a card from a pack of shuffled cards:
A pack or deck of playing cards has 52 cards which are divided into four categories as given below
Spades (♠)
Clubs (♣)
Hearts (♥)
Diamonds (♦)
Each of the above-mentioned categories has 13 cards, 9 cards numbered from 2 to 10, an Ace, a King, a Queen, and a jack
Hearts and Diamonds are red-faced cards whereas Spades and Clubs are black faced cards. Kings, Queens, and Jacks are called face cards
Taking a ball randomly from a bag containing balls of different colors.
Events
The possible outcomes of a trial are called events. So, any subset of a sample space S is called an event. For example, (i) when a coin is tossed, the outcome of getting a head or tail is an event. (ii) When a die is thrown, the outcome of getting 1 or 2 or 3 or 4 or 5 or 6 is an event.
Equally likely events: The events are said to be equally likely if there is no reason to expect anyone in preference to any other. For example, (i) When a dice is thrown, then all the six faces (1, 2, 3, 4, 5, 6) are equally likely to come. (ii) When a card is drawn from a well-shuffled pack, then all the 52 cases are equally likely to come.
Exhaustive events: It is the total number of all possible outcomes of any trial. For example, (i)When a coin is tossed, we get either Head or Tail. Hence there are 2 exhaustive events. (ii)When a dice is thrown, there are 6 exhaustive events.
Mutually Exclusive events: Two or more events are said to be mutually exclusive events if they cannot happen simultaneously in the trial. For example, (i) When a coin is tossed, either Head or Tail will appear. Head and Tail cannot come simultaneously. Hence occurrence of Head and Tail are mutually exclusive events. (ii) When a die is thrown, any one of the six faces will appear.
Favorable events: The cases, which ensure the occurrence of the events, are called. For example, (i) When two dice are thrown, the number of cases favorable for getting a sum 6 is 5 viz (1,5), (5,1), (2,4), (4,2), (3,3).
Independent and dependent events: Events are said to be independent if the occurrence or non-occurrence of one event does not influence the occurrence or non-occurrence of any other, otherwise they are said to be dependent. For example, When a card is drawn from a pack of well-shuffled cards and replaced before drawing the second card, the result of the second draw is independent of the first one. However, if the first card is not replaced, the second draw is dependent on the first one.
Simple and Compound events: If an event has only one sample point of a sample space, it is called a simple (elementary) event. If an event has more than one sample point, it is called a compound event. For example, When we toss two coins, (i) for sample space {HT, TH, HH, TT}, then sample point {TT} of getting both tails is a simple event. (ii) for a subset of sample space {HT, TH, TT} of getting at least one tail is a compound event.
Algebra of Events
Let A, B be events associated with an experiment whose sample space is S.
Complimentary 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, in throwing a dice, the sample space is {1,2,3,4,5,6}. E represents an event of getting an even number, i.e. E={2,4,6}. Then event E’ represents the event of not even number, i.e E’={1,3,5}.
Event A or B: The event A or B shows the sample points of a random experiment which are either in A or B or both. This event is represented by ‘A U B’. Suppose A = {2,3,5} and B={1,3,5}, then A U B = {1,2,3,5}.
Event A and B: The event A and B show the sample points of a random experiment which are common to both A and B. This event is represented by ‘A ∩ B’. Suppose A = {2,3,5} and B={1,3,5}, then A ∩ B = {3,5}.
Event A but not B: The event A but not B shows the sample points which are in A but not in B. Event A but not B is represented by A ∩ B’ = A – A ∩ B. Suppose event A = {1, 3, 4, 5, 6, 7} and B’ = {2, 3, 5. 6}. A ∩ B’ = {1, 4, 7}.
The probability of an event
Let E be an event and S be the sample space. The probability of the event E can be defined as
P(E) = n(E) / n(S)
where P(E) = Probability of the event E, n(E) = number of ways in which the event can occur and n(S) = Total number of outcomes possible. For example, a coin is tossed, sample space is {H, T}. So, n(S) = 2. E = event of getting tail, so n(E) = 1.
P(E) = n(E) / n(S) = ½.
*NOTE : "This study material is collected from multiple sources to make a quick refresh course available to students."