How can I solve any question on finding the probability of an event in a job interview / written test? (Part 1)

Whenever you face probability questions, you find that the questions are very interesting but the answers are confusing. In many cases, you find that your intuition goes wrong and several times you find no clue how to get the correct answer. In this post, I will discuss how you can systematically arrive to the solution of any question related to probability calculation. To begin with, we will discuss some of the key terms related to probability. 

Random Experiment (E): An experiment or process whose all possible outcomes are known in advance but we do not know which outcome will appear in a particular performance of the experiment. 

Sample Space (S): Set of all outcomes related to a random experiment E. Each element of S is called a sample point.  

Example 1: Consider a random experiment E1= tossing of a coin. Here we know beforehand that in each performance of E1, either head (H) or tail (T) will appear. But we do not know which one will certainly appear.  Therefore, E1 is a random experiment and the corresponding sample space is S1 = {H,T}. H and T are called sample point of E1.

Example 2: Consider E2= Tossing of two coins or tossing a coin two times. Here E2 = E1 × E1, repeating E1 two times. The sample space of E2 will be S2 = S1×S1 = {H,T}×{H,T} = {HH, TH, HT, TT}. Note that each outcome (sample point) of E2, is a combination of two outcomes of tossing of a coin, the 1st one corresponding to 1st coin toss and the 2nd one corresponding to 2nd coin toss.  The sample points are HH, TH, HT and TT. 

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Event (A): An event A of a random experiment E, is nothing but a subset of the sample space S.  When A = ɸ (nothing), it is called null event and when A = S (the whole sample space), it is called the certain event. 

Example 1 continue: In this example the sample space S1 = {H,T} contains two elements (sample points). So, there exist 4 possible subsets (events) of S1 and they are ɸ (null event), {H} (the head), {T} (the tail) and the certain event {H,T} (either head or tail). 

Example 2 continue: In this example the sample space S2 = {HH, TH, HT, TT} contains four elements (sample points). So, there exist 16 possible subsets (events) of S2 and some of them are like: 

A = ɸ; ‘nothing’, 
A = {HH}; ‘both head’,
A= {TH, HT}; ‘exactly one head’ or ‘exactly one tail’ or ‘both head and tail’
A = {HH, TH, HT}; ‘at least one head’ or ‘at most one tail’. 

Probability(P): The probability P corresponding to a random experiment E, is a rule that assigns a value between 0 to 1, to each event A.

We will now discuss some useful properties of probability that are frequently used in probability calculations. 


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