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Probability: Introduction, examples, MCQ Quizzes- Conditional, Compound Probability; Random Variables; Baye's Theorem

Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE, Anyone else who needs this Tutorial as a reference!

General Outline of the ideas introduced in this tutorial

Introductory Probability- Compound and Independent Events, Mutually Exclusive Events, Multi-Stage Experiments

• On the most basic level, probability is defined as : P(x)=number of favourable outcomes/total number of outcomes
• Here, P(x) is the probability of occurrence of an event x, and the event x here is a discrete and independent event.
• The definition of probability in general can be for continuous events, dependent events, conditional events and so on.
• In probability theory, we perform or consider the experiments in which all possible outcomes are known in advance.
• P(x)+P(x')=1 ;that is, sum of probability of an event occurring and the probability of the same event not occurring is one.
• Compound Events are formed by combining two or more simple events:
• A∩B: A intersection B. A and B happening together, simultaneously.
• A∪B : A union B. Favorable outcomes are those which are favorable to A or B or both.
• P(A∪B)=P(A)+P(B)-P(A∩B)
• Two events A and B are said to be Mutually Exclusive Events if P(A∩B)=0, that is, if both A and B can NEVER happen simultaneously.
• Two events are said to be Independent, if P(A|B)=P(A), or the conditional probability of event A occurring given event B has occurred is equal to the probability of event A, or that event A is unaffected by event B
• Conditional Probability: Conditional Probability is defined as the probability of occurring of an event given that some other events have already occurred.  Let A and B be such events. Then probability of A occurring given that B has occurred is P(A|B)=P(A∩B)P(B)
• If A and B are independent then, P(A∩B)=P(A)P(B)
• An experiment can be single stage or multi-stage, depending upon if an experiment can be further broken down into stages.
• In case of a multi-stage experiment, we can have conditional probabilities between events A and B such that either :
• A and B are in the same stage. In this case we use: P(AB)=n(A∩B)n(B)
• A is in the stage that has occurred, and B is in a stage that will occur (experiment is not yet over).
• A is in a previous stage, and B is in a later stage, and the experiment is over.
• In the second and third case, we make use of Baye’s network and Baye’s theorem

Application of Probability and a quick example of a Probability Distribution, Random Variables and Sample Space

• Probability finds its application in a wide variety of areas. Be it Artificial Intelligence, communication, statistical analysis, quantum mechanics and many more.
• One way of using probability is introducing Random variables. If S is the sample space, and w is an element in sample space, then a real valued function X which assigns a unique real number X(w) to each w in S is called a random variable X.
• Example: Let a coin be tossed twice. S={HH,HT,TH,TT}
Let X be the random variable defined as :
X:S->R:X(w)=number of tails in w, for all w in S.
Thus, X(HH)=0,X(TH)=X(HT)=1,X(TT)=2.
• There can be a probability distribution associated with a random variable. If a random variable X takes values x1,x2,x3…,xn with probabilities p1, p2, p3, …, pn, then probability distribution is given by the following table. Each pi>=0 and Sum of all p
 X x1 x2 x3 … … xn P(X) p1 p2 p3 … … pn

Here are some of the problems covered in this tutorial :

Q: If (1-3p)/2, (1+4p)/3, (1+p)/6 are the probabilities of three mutually exclusive and exhaustive events, then find the value(s) of p.

Q: What is the probability of a leap year having 53 Saturdays or 53 Sundays?

Q: A box contains 10 good articles and 6 defective articles. One item is drawn at random. What is the probability that it is either good or has a defect?

Q: The probabilities that a student will obtain grades A,B,C or D are 0.30,0.35,0.20 and 0.15 respectively. What is the probability that the student will receive atleast grade C?

Q: If the probability of A failing in an exam is 0.2 and of B failing is 0.3, then what is the probability of either A failing, or B failing?

Q: A bag contains 7 red and 2 white balls and another bag contains 5 red and 4 white balls. Two balls are drawn, one from each bag. What is the probability that both balls are white?

Q: Two athletes A and B participate in a race along with other athletes. If the chance of A winning the race is 1/16 and that of B winning the same race is 1/8, what is the chance that neither wins the race?

Q: Three integers are chosen at random from first twenty integers. The probability that their product is even is?(NOTE: For this question, basic Permutation and Combination knowledge is required)

Q: An integer is chosen randomly from the numbers 1,2,…,25. What is the probability that the chosen number is divisible by 3 or 4?

Q: The probability that at least one of the events A and B occurs is 0.6. If A and B occur simultaneously with probabilities 0.2, then P(not A)+P(not B) = ?

Q: A bag contains x white and y black balls. Two players A and B alternately draw a ball from the bag and then replace it every time after the draw. A begins the game. If the probability of A winning (A player wins when he draws a white ball) is twice the probability of B winning, then find x:y.

Q: A fair coin is tossed n times. Let X=number of times head occurs. If P(X=4), P(X=5) and P(X=6) are in AP, find the value(s) of n.

Q: A person writes four letters and addresses four envelopes. If the letters are placed in the envelopes at random, then what is the probability that all the letters are not placed in the correct envelopes?

Q: A card is drawn from a pack of 52 cards. Let the events be defined as follows:
A : card drawn is red.
B : card drawn is an ace.
C : card drawn is a spade.
Find P(A/B), P(B/C),P(C/A),P(A/C).

Q: Two balls are drawn without replacement one after the other from a bag having 5 white and 8 black balls. What are the chances that the second ball drawn is black?

Q: Let probability that a person has cancer be 0.01. There are two tests T1 and T2. Probability of a test coming positive given that person had cancer is 0.9. Probability of a test being negative given that person did not have cancer is 0.8. What is the probability that a person has cancer given that both T1 and T2 came out positive?

Q: In a game, 3 coins are tossed. A person is paid 5 rupees if he gets all heads or all tails, and he is supposed to pay 3 rupees if gets one head or two heads. What can he expect to win on an average per game?

Q: A number x is selected from first 100 natural numbers. Find the probability that x satisfy the condition x + 100/x > 50

Q: There are three events A, B and C one of which must, and only one can happen, the odd are 8 to 3 against and 2 to 5 for B. Find the odd, against C.

Complete Tutorial with Problems and Solutions :

MCQ Quiz #1 on Basic Probability

Companion MCQ Quiz #1 for Probability- test how much you know about the topic. Your score will be e-mailed to you at the address you provide.