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Probability
The theory of probability originated from the game of chance and gambling. In days of old, gamblers used to gamble in a gambling house with a die to win the amount fixed among themselves. They were always desirous to get the prescribed number on the upper face of a die when it was thrown on a board. Shakuni of Mahabharat was perhaps one of them. People started to study the subject of probability from the middle of seventeenth century. The mathematicians Huygens, Pascal Fermat and Bernoulli contributed a lot to this branch of Mathematics. A.N. Kolmogorow proposed the set theoretic model to the theory of probability.
characteristics constant of the radioactive element. The mathematician Boltzmann in his theorem in classical statistics related entropy to probability. The application of probability theory in quantum mechanics is of greater importance. Here the probability of finding an electron in a state with certain quantum numbers in an element of given volume in the vicinity of a point is determined. This branch of mathematics has applications in applied sciences and statistics also when we plan and organize production. The methods of this theory help in solving the diverse problems of natural science as a result of which its study becomes essential now.
Probability gives us a measure for likelihood that something will happen. However it must be appreciated that probability can never predict the number of times that on occurrence actually happens. But being able to quantify the likely occurrence of an event is important because most of the decisions that affect our daily lives are based on likelihoods and not on absolute certainties.
DEFINITIONS
In every case it is set of some or all possible outcomes of the experiment. Therefore event (A) is subset of sample space (S). If outcome of an experiment is an element of A we say that event A has occurred.
4. Complement of an event : The set of all out comes which are in S but not in A is called the Complement of the Event A
denoted by , A^{c },A’ or not A’
5. Compound Event : if A and B are two given events than A ∩ B is called Compound Event and is denoted by A ∩ B or AB
or A and B
6. Mutually Exclusive Events : Two events are said to be Mutually Exclusive (or disjoint or incompatible) if the occurrence
of one precludes (rules out) the simultaneous occurrence of the other if A and B are two mutually exclusive events then
P(A and B) = 0
Consider for example, choosing numbers at random from the set {3,4,5,6,7, 8,9,10,11,12} if , Event A is the selection of
a prime number Event B is the selection of an odd number, Event C is the selection of an odd number, then A and C are
mutually exclusive as none of the numbers in this set is both prime and even. But A and B are not mutually exclusive as
some numbers are both prime and odd (viz. 3,5,7,11)
7. Equally Likely Events : Events are said to be Equally Likely when each event is as likely to occur as any other event.
8. Exhastive Events : Events A,B,C ………..L are said to be Exhastive Event if no event outside this set can result as an
outcome of an experiment For example, if A and B are two events defined on a sample space S, then A and B are
exhaustive
CLASSICAL DEFINITION OF PROBABILITY
If n represents the total number of equally likely, mutually exclusive and exhaustive outcomes of an experiment and m of them are favourable to the happening of the event A, then the probability of happening of the event A is given by P(A) = m/n.
Comparitive study of Equally likely. Mutually Exclusive and Exhasutive events :
Experiment |
Events |
E/L |
M/E |
Exhaustive |
1. Throwing of a die |
A:throwing an odd face {1,3,5} B:throwing a composite {4,6} |
No |
Yes |
No |
2. A ball is drawn from an um containing2w,3R and 4G balls |
E_{1} : getting a W ball E_{2} : getting a R ball E_{3}: getting a G ball |
No |
Yes |
Yes |
3. Throwing a pair of dice |
A:Throwing a doublet {11,22,33,44,55,66} |
Yes |
No |
No |
4. From a well shuffled pack of cards a card is drawn |
E_{1} : getting a heart E_{2} : getting a space E_{3}: getting a diamond E_{4 }: getting a club |
Yes |
Yes |
Yes |
5. From a well shuffled pack of a cards a card is drawn |
A = getting a heart B = getting a face card |
No |
No |
No |
THE PROBABILITY THAT AN EVENT DOES NOT HAPPEN
If, in a possibility space of n equally likely occurences the number of times an event A occurs is r, there are n-r occasions when A does not happen The event A does not happen ‘ is denoted by ( and is read as ‘not A’)
This relationship is most useful in the ‘at least one’ type of problem as is illustrated below.
Example : if four cards are drawn at random from a pack of fifty-two playing cards find the proabability that at least one of them is an ace.
If A is a combination of four cards containing at least one ace. (i.e., either one ace, or two aces, or three aces or four aces) them is combination of four cards containing no aces
SECTION -B
VENN DIAGRAM BASED PROBLEMS
VENN DIAGRAMS
A diagram used to illustrate relationships between sets. Commonly, a rectangle represents the universal set and a circle within it represents a given set (all members of the given set are represented by points within the circle). A subset is represented by a circle within a circle and union and intersection are indicated by overlapping circles. Let S is the sample space of an experiment and A, B, C are three events corresponding to it
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