Bayes' Theorem

In medical diagnostics for a disease, sensitivity (denoted a) of a test refers to the probability that a test result is positive for a person with the disease whereas specificity (denoted b) refers to the probability that a person without the disease test negative. A diagnostic test for influenza has the values of $\mathrm{a}=0.9$ and $\mathrm{b}=0.9.$ Assume that the prevalence of influenza in a population in $50\%.$ If a randomly chosen person tests negative, what is the probability that the person actually has influenza?

In medical diagnostics for a disease, sensitivity (denoted by $a$) of a test refers to the probability that a test result is positive for a person with the disease, whereas specificity (denoted by $b$) refers to the probability that a person without the disease tests negative. A diagnostic test for $\mathrm{COVID}-19$ has the values of $a=0.99$ and $b=0.99$. If the prevalence of $\mathrm{COVID}-19$ in a population is estimated to be $10\%$, what is the probability that a randomly chosen person tests positive for $\mathrm{COVID}-19$ ?

The probability of men getting a certain disease is $\frac{1}{2}$ and that of women getting the same disease is $\frac{1}{5}.$ The blood test that identifies the disease gives the correct result with probability $\frac{4}{5}$. Suppose a person is chosen at random from a group of $30$ males and $20$ females, and the blood test of that person is found to be positive. What is the probability that the chosen person is a man?

Ravi and Rashmi are each holding $2$ red cards and $2$ black cards (all four red and all four black cards are identical). Ravi picks a card at random from Rashmi and then Rashmi picks a card to random from Ravi. This process is repeated a second time. Let $p$ be the probability that both have all $4$ cards of the same colour. Then, $p$ satisfies

In a certain recruitment test with multiple choice questions, there are four options to each question, out of which only one is correct. An intelligent student knows $90\%$ of the correct answers while a weak student knows only $20\%$of the correct answers. If a weak student gets the correct answer, the probability that he was guessing is

An insurance company insured $2000$ scooters and $3000$ motorcycles. The probability of an accident involving a scooter is $0.01$, and that of motorcycles is $0.02$. An insured vehicle met with an accident. Find the probability that the accidented vehicle was a motorcycle.

In a bulb factory, machines $\mathrm{A}, \mathrm{B}$ and $C$ manufactures $60\%, 30\%$ and $10\%$ bulbs respectively. Out of these bulbs $1\%, 2\%$ and $3\%$ of the bulbs produced respectively by $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are found to be defective. A bulb is picked up at random from the total production and found to be defective. Find the probability that this bulb was produced by machine $\mathrm{A}$.

A manufacturer has three machine operators $A, B$ and $C$. The first operator $A$ produces $1\%$, whereas the other two operators $B$ and $C$ produce $5\%$ and $1\%$ defective items respectively. $A$ is on the job for $50\%$ of the time, $B$ is on the job for $30\%$ of the time and $C$ is on the job for $20\%$ of the time. A defective item is produced, what is the probability that it was produced by $A$?

A letter comes from the two cities $\mathrm{TATANAGAR}$ or $\mathrm{CALCUTTA}$. Only alphabets $\mathrm{TA}$ is visible on the envelope. Find the probability that the letter comes from the city $\mathrm{TATANAGAR}$.

On a multiple choice examination with four choices a student either guesses or knows or cheat the answer.The probability that he makes a guess is $\frac{1}{3}$and probability that he copies the answer is $\frac{1}{6}$.The probability that his answer is correct given that he copies it is$\frac{1}{8}$.Find the probability of guessing or cheating the answer if it is known that he answers the question correctly.

A person has undertaken a construction job. The probabilities are $0.65$ that there will be strike, $0.80$ that the construction job will be completed on time if there is no strike and $0.32$ that the construction job will be completed on time with strike, then find the probability that construction job will be completed on time.

A card from a pack of $52$ cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.

Suppose that $5\%$ of men and $0.25\%$ of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.

Students in a college, it is known that $60\%$ reside in hostel and $40\%$ are day scholar (not residing in hostel). Previous year results report that $30\%$ of all students who reside in hostel attain A grade and $20\%$ of day scholar attain A grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is a hosteler ?

Three urns are given, each containing red and black balls as indicated below:

An urn is chosen at random and a ball is drawn from that urn. The ball drawn is red. The probability that the ball is drawn either from urn $\mathrm{II}$ or from urn $\mathrm{III}$ is given by $\frac{x}{y}$. Find $x+y$.

There are two packs $A$ and $B$ of $52$ playing cards. All the four aces from the pack $A$ are removed whereas from the pack $B$, one ace, one king, one queen and one jack is removed. One of these two packs is selected at random and two cards are drawn simultaneously from it and found to be a pair (i.e., both have same rank e.g. two $9$ 's or two queen etc). If $\frac{m}{n}$ (expressed in lowest form) is the probability that the pack $A$ was selected, then find $\left(m+n\right)$.

Suppose $5\%$ of men and $0.25\%$ of women have grey hair. A grey-haired person is selected at random. What is the probability of this person being male? Assume that there is an equal number of males and females.

In a class, $5\%$ of the boys and $10\%$ of the girls have an$IQ$ of more than $150$. In this class, $60\%$ of the students are boys. If a student is selected at random and found to have $IQ$ of more than $150$, find the probability that the student is a boy.

In a certain college, $4\%$ of boys and $1\%$ of girls are taller than $1.75 \mathrm{meters}$. Furthermore, $60\%$ of the students are girls. If a student is selected at random and is taller than $1.75 \mathrm{meters}$, what is the probability that the selected student is a girl?