Definitions [10]
The conditional probability of both events A and B over the sample space S is
\[\mathrm{P(A/B)=\frac{P(A\cap B)}{P(B)}}\], where \[B\neq\phi\]
\[\mathrm{P(B/A)=\frac{P(A\cap B)}{P(A)}}\], where \[A\neq\phi\]
Two events are said to be independent if the occurrence of one does not depend on the other.
If A and B are independent events, then
- P(A/B) = P(A/B') = P(A)
- P(B/A) = P(B/A') = P(B)
- If A and B are independent events, then
a. P(A∩ B) = P(A). P (B)
b. A and B' are also independent
c. A' and B' are also independent
The conditional event of A given B means the occurrence of A under the condition that B has already occurred.
It is denoted by A|B.
If a random variable x can take values x1, x2,…, xn with probabilities p(x1) ,p(x2),…, p(xn) such that p(x1) + p(x2) +… + p(xn) = 1, the function p is called the probability density function of x and is said to define the probability distribution of x.
Random variable:
A random variable is a variable whose values depend on chance and are the result of a random observation or experiment.
Discrete random variable:
If the set of values taken by a random variable can be counted and listed, it is called a discrete random variable.
Continuous Random Variable:
If the set of values is continuous, the variable is called a continuous random variable.
Let A and B be two events associated with a random experiment. Then, the probability ofthe occurrence of A under the condition that B has already occurred and P(B) ≠ 0, is called the conditional probability of A given B and is written as P(A/B).
Trials of a random experiment are called Bernoulli’s trials if they satisfy the following conditions:
-
The number of trials is finite.
-
Each trial is independent of the others.
-
Each trial has exactly two outcomes: success or failure.
-
The probability of success (or failure) remains the same in each trial.
Statement:
Let p be the probability of success of an event and q be the probability of failure of the event in one trial. Suppose there are n trials of the event in a binomial experiment, then the binomial probability distribution is defined by the following table:
| Number of successes X | 0 | 1 | 2 | ...r | ...n |
|---|---|---|---|---|---|
| Probability P(X) | qn | nC1pqn−1 | nC2p2qn−2 | ...nCrprqn−r | ...pn |
Mean µ (Greek mu) of the above probability distribution may be defined as
\[\mu=\frac{p_1x_1+p_2x_2+p_3x_3+.......+p_nx_n}{p_1+p_2+p_3+......+p_n}\]
\[=\frac{\sum p_ix_i}{\sum p_i}=\Sigma p_ix_i\]
\[Mean\overline{x}=\sum_{i=1}^{n}p_{i}x_{i}\],where each pi \[P_{i}\geq0\] and \[\sum p_{i}=p_{1}+p_{2}+...+p_{n}=1\]
Independent events:
A set of events is said to be independent if the occurrence of any one of them does not, in any way, affect the occurrence of any other in the set.
Dependent events:
Two events E and F are said to be dependent if they are not independent, i.e. if \[\mathrm{P}(\mathrm{E}\cap\mathrm{F})\neq\mathrm{P}(\mathrm{E}).\mathrm{P}(\mathrm{F})\]
Formulae [8]
\[\begin{gathered}
\mu=\int_{-\infty}^{\infty}xf(x)dx \\
\sigma^2=\int_{-\infty}^\infty(x-\mu)^2f(x)dx
\end{gathered}\]
If n = 2:
\[P(E_1\mid A)=\frac{P(E_1)P(A\mid E_1)}{P(E_1)P(A\mid E_1)+P(E_2)P(A\mid E_2)}\]
\[P(E_2\mid A)=\frac{P(E_2)P(A\mid E_2)}{P(E_1)P(A\mid E_1)+P(E_2)P(A\mid E_2)}\]
Bayes’ theorem for three events:
\[P(E_1\mid A)=\frac{P(E_1)P(A\mid E_1)}{P(E_1)P(A\mid E_1)+P(E_2)P(A\mid E_2)+P(E_3)P(A\mid E_3)}\]
| Concept | Mathematical Form | Important Condition |
|---|---|---|
| Conditional Probability of A given B | \[P(A\mid B)=\frac{P(A\cap B)}{P(B)}\] | P(B) ≠ 0 |
| Conditional Probability of B given A | \[P(B\mid A)=\frac{P(A\cap B)}{P(A)}\] | P(A) ≠0 |
The variance of a random variable x is denoted by σ2.
First form: \[\sigma^2=\sum_{i=1}^np_i(x_i-\mu)^2\]
Second form: \[\sigma^2=\sum_{i=1}^np_ix_i^2-\mu^2\]
\[P(A\cap B)=P(A).P(B/A)=P(B).P(A/B)\]
Extension of Multiplication Theorem:
\[P(A\cap B\cap C)=P(A)P(B/A).P(C/A\cap B)\]
Mean: μ = np
Variance: σ2 = npq
Standard deviation: \[\sigma=\sqrt{npq}\]
\[\sigma=\sqrt{\sigma^2}=\sqrt{\sum p_ix_i^2-\mu^2}\]
General Form: \[P(X=r)={}^nC_rp^rq^{n-r},\quad r=0,1,2,\ldots,n\]
Theorems and Laws [4]
If A and B are two events over the sample space S, then
- P(A ∩ B) = P(B) · P (A/B)
- P(A ∩ B) = P(A) · P (B/A)
If B1, B2,..., Bn are mutually exclusive and exhaustive events and if A is an event consequent to these Bi's, then for each i = 1, 2, 3, ..., n,
\[\mathrm{P(B_i/A)=\frac{P(B_i)P(A/B_i)}{\sum_{i=1}^nP(A\cap B_i)}}\]
Statement:
Let S be the sample space and E1, E2,…, En be mutually exclusive and exhaustive events associated with a random experiment. Let A be any event associated with S. Then,
\[P(A)=P(E_1)P(A\mid E_1)+P(E_2)P(A\mid E_2)+\cdots+P(E_n)P(A\mid E_n)\]
or
\[P(A)=\sum P(E_i)P(A\mid E_i)\]
Statement:
Let E1, E2,…, En be mutually exclusive and exhaustive events, and A be an event such that P(A) ≠ 0. Then,
\[P(E_i\mid A)=\frac{P(E_i\cap A)}{P(E_1\cap A)+P(E_2\cap A)+\cdots+P(E_n\cap A)}\]
Second form:
\[P(E_i\mid A)=\frac{P(E_i)P(A\mid E_i)}{\sum P(E_j)P(A\mid E_j)}\]
Key Points
| Step | What to do | form |
|---|---|---|
| 1 | Find the probability of the first event | P(A) |
| 2 | Find the probability of the second event after the first | P(B|A) |
| 3 | Multiply | \[P(A\cap B)=P(A)P(B\mid A)\] |
| Type | Meaning |
|---|---|
| Prior probabilities | \[P(E_1),P(E_2),\ldots,P(E_n)\] |
| Likelihood probabilities | \[P(A\mid E_1),P(A\mid E_2),\ldots\] |
| Posterior probabilities | \[P(E_1\mid A),P(E_2\mid A),\ldots\] |
| Type of Event | Meaning / Condition | Probability Formula |
|---|---|---|
| Simple Event | Single outcome | \[P(A)=\frac{\text{favourable}}{\mathrm{total}}\] |
| Compound Event | More than one outcome | Depends on the situation |
| Mutually Exclusive Events | Cannot occur together | \[P(A\cup B)=P(A)+P(B)\] |
| Not Mutually Exclusive (Inclusive) | Can occur together | \[P(A\cup B)=P(A)+P(B)-P(A\cap B)\] |
| Exhaustive Events | Cover the entire sample space | \[P(A\cup B)=1\] |
| Complementary Events | One is NOT the other | \[P(A^{\prime})=1-P(A)\] |
| Event & Complement | Cannot occur together | P(A) + P(A') = 1 |
| At least one of A or B | A or B or both | \[P(A\cup B)\] |
| Neither A nor B | Neither occurs | \[P(A^{\prime}\cap B^{\prime})=1-P(A\cup B)\] |
| Breaking Event A | Using B & B′ | \[P(A)=P(A\cap B)+P(A\cap B^{\prime})\] |
| Breaking Event B | Using A & A′ | \[P(B)=P(A\cap B)+P(A^{\prime}\cap B)\] |
-
Probabilities are terms of (q + p)n.
-
P(0) + P(1) + ⋯ + P(n) = 1.
-
The binomial distribution is discrete.
-
n and p are its parameters.
Special cases:
-
P(0) = qn
-
P(1) = npqn−1
Important Questions [41]
- If for any two events A and B, P(A) = 45 and P(A ∩ B) = 710, then P(BA) is equal to ______.
- If for two events A and B, P(A – B) = 15 and P(A) = 35, then P(BA) is equal to ______.
- A Die is Thrown Three Times. Events a and B Are Defined as Below: a : 5 on the First and 6 on the Second Throw. B: 3 Or 4 on the Third Throw. Find the Probability of B, Given that a Has Already Occurred.
- If A and B are two events such that P(AB)=2×P(BA) and P(A) + P(B) = 23, then P(B) is equal to ______.
- If the sum of numbers obtained on throwing a pair of dice is 9, then the probability that number obtained on one of the dice is 4, is ______.
- 40% Students of a College Reside in Hostel and the Remaining Reside Outside. at the End of the Year, 50% of the Hostelers Got a Grade While from Outside Students, Only 30% Got a Grade in the Examination.
- A Box Has 20 Pens of Which 2 Are Defective. Calculate the Probability that Out of 5 Pens Drawn One by One with Replacement, at Most 2 Are Defective.
- Three Cards Are Drawn at Random (Without Replacement) from a Well-shuffled Pack of 52 Playing Cards. Find the Probability Distribution of the Number of Red Cards.
- A Bag X Contains 4 White Balls and 2 Black Balls, While Another Bag Y Contains 3 White Balls and 3 Black Balls. Two Balls Are Drawn (Without Replacement) at Random from One of the Bags and Were Found to Be One White and One Black.
- If A and B are two independent events such that P(A) = 13 and P(B) = 14, then P(B′A) is ______.
- Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls? Given that
- Read the following passage: Recent studies suggest the roughly 12% of the world population is left-handed.Depending upon the parents, the chances of having a left-handed child are as follows:
- In a Game, a Man Wins Rs 5 for Getting a Number Greater than 4 and Loses Rs 1 Otherwise, When a Fair Die is Thrown. the Man Decided to Thrown a Die Thrice but to Quit as and When He Gets a Number Greater than 4. Find the Expected Value of the Amount He Wins/Loses
- Determine P(E|F). Mother, father and son line up at random for a family picture E: son on one end, F: father in middle
- A black and a red dice are rolled. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.
- Assume that the chances of a patient having a heart attack is 40%. Assuming that a meditation and yoga course reduces the risk of heart attack by 30% and prescription of certain drug reduces its chance by 25%. At a time a patient can choose any one of the two options with equal probabilities.
- If A and B are two independent events such that P(A∩ B) =2/15 and P(A ∩ B) = 1/6, then find P(A) and P(B).
- A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards are drawn at random (without replacement) and are found to be all spades. Find the probability of the lost card being a spade.
- A speaks truth in 60% of the cases, while B in 90% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact?
- A bag contains 4 balls. Two balls are drawn at random (without replacement) and are found to be white. What is the probability that all balls in the bag are white?
- A Die, Whose Faces Are Marked 1, 2, 3 in Red and 4, 5, 6 in Green is Tossed. Let a Be the Event "Number Obtained is Even" and B Be the Event "Number Obtained is Red". Find If a and B Are Independent
- A fair coin and an unbiased die are tossed. Let A be the event ‘head appears on the coin’ and B be the event ‘3 on the die’. Check whether A and B are independent events or not.
- Prove that If E and F Are Independent Events, Then the Events E and F' Are Also Independent.
- If P(A) = 0·4, P(B) = P, P(A ⋃ B) = 0·6 and a and B Are Given to Be Independent Events, Find the Value of 'P'.
- The probabilities of solving a specific problem independently by A and B are 1/3 and 1/5 respectively. If both try to solve the problem independently, find the probability that the problem is solved.
- Events A and Bare such that P(A) = 12, P(B) = 712 and P(A¯∪B¯)=14. Find whether the events A and B are independent or not.
- The probability that A hits the target is 13 and the probability that B hits it, is 25. If both try to hit the target independently, find the probability that the target is hit.
- Five fair coins are tossed simultaneously. The probability of the events that at least one head comes up is ______.
- Three persons A, B and C apply for a job a manager in a private company. Chances of their selection are in the ratio 1:2:4. The probability that A
- In a factory, machine A produces 30% of total output, machine B produces 25% and the machine C produces the remaining output. The defective items produced by machines A, B
- There are two boxes, namely box-I and box-II. Box-I contains 3 red and 6 black balls. Box-II contains 5 red and 5 black balls. One of the two boxes, is selected at random
- In answering a question on a multiple choice test, a student either knows the answer or guesses. Let 3/5 be the probability that he knows the answer and 2/5 be the probability that he guesses.
- Read the following passage and answer the questions given below. A shopkeeper sells three types of flower seeds A1, A2, A3. They are sold is the form of a mixture
- Suppose a Girl Throws a Die. If She Gets 1 Or 2 She Tosses a Coin Three Times and Notes the Number of Tails. If She Gets 3,4,5 Or 6, She Tosses a Coin Once and Notes Whether a ‘Head’
- Often It is Taken that a Truthful Person Commands, More Respect in the Society. a Man is Known to Speak the Truth 4 Out of 5 Times. He Throws a Die and Reports that It is a Six. Find the Probability that It is Actually a Six
- Of the Students in a School, It is Known that 30% Have 100% Attendance and 70% Students Are Irregular at the End of the Year, One Student is Chos~N at Random from the School and He Was Found ·To Have an a Grade. What is the Probability that the Student Has 100% Attendance? is Regularity Required Only in School? Justify Your Answer
- A Manufacturer Has Three Machine Operators A, B and C. the First Operator a Produces 1% Defective Items, Where as the Other Two Operators B and C Produce 5% and 7%
- Three Machines E1, E2 and E3 in a Certain Factory Producing Electric Bulbs, Produce 50%, 25% and 25% Respectively, of the Total Daily Output of Electric Bulbs.
- An Insurance Company Insured 2000 Scooter Drivers, 4000 Car Drivers and 6000 Truck Drivers. the Probabilities of an Accident for Them Are 0.01, 0.03 and 0.15, Respectively.
- Three persons A, B and C apply for a job of Manager in a Private Company. Chances of their selection (A, B and C) are in the ratio 1 : 2 :4. The probabilities that A, B and C can introduce changes to improve profits of the company are 0.8, 0.5 and 0.3, respectively. If the change does not take place, find the probability that it is due to the appointment of C
- There are three coins. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the times and the third is also a biased coin that comes up tails 40% of the time. One of the three coins is chosen at random and tossed and it shows heads. What is the probability that it was the two-headed coin?
