Maharashtra Board Solutions for Chapter: Probability, Exercise 4: EXERCISE 9.4

A box contains $2$ blue and $3$ pink balls and another box contains $4$ blue and $5$ pink balls. One ball is drawn at random from one of the two boxes and it is found to be pink. Find the probability that it was drawn from 

$\left(\mathrm{i}\right)$ First box

$\left(\mathrm{ii}\right)$ Second box.

If $E_1$ and $E_2$ are equally likely, mutually exclusive and exhaustive events and $P\left(A/E_1\right)=0.2,P\left(A/E_2\right)=0.3$ Find $P\left(E_1/A\right)$.

A diagnostic test has a probability $0.95$ of giving a positive result when applied to a person suffering from a certain disease, and  a probability $0.10$ of giving a (false) positive result when applied to a non-sufferer. It is estimated that $0.5\%$ of the population are sufferers. Suppose that the test is now administered to a person about whom we have no relevant information relating to the disease (apart from the fact that he/she comes from this population). Calculate the probability that given a negative result, the person is a non-sufferer.

A doctor is called to see a sick child. The doctor has prior information that $80\%$of the sick children in that area have the flu, while the other $20\%$are sick with measles. Assume that there is no other disease in that area. A well-known symptom of measles is rash. From the past records, it is known that, chances of having rashes given that sick child is suffering from measles is $0.95$. However occasionally children with flu also develop rash, whose chance are $0.08$. Upon examining the child, the doctor finds a rash. What is the probability that child is suffering from measles?

$2\%$ of the population have a certain blood disease of a serious form: $10\%$ have it in a mild form; and $88\%$ don't have it at all. A new blood test is developed; the probability of testing positive is $\frac{9}{10}$ if the subject has the serious form, $\frac{6}{10}$ if the subject has the mild form, and $\frac{1}{10}$if the subject doesn't have the disease. A subject is tested positive. What is the probability that the subject has serious form of the disease?

A box contains three coins: two fair coins and one fake two-headed coin is picked randomly from the box and tossed. What is the probability that it lands head up?

A box contains three coins: two fair coins and one fake two-headed coin is picked randomly from the box and tossed.

If happens to be head, what is the probability that it is the two-headed coin?

There are three social media groups on a mobile: Group $I$, Group $II$ and Group $III$. The probabilities that Group $I$, Group $II$ and Group $III$ sending the messages on sports are $\frac{2}{5},\frac{1}{2},$ and $\frac{2}{3}$ respectively. The probability of opening the messages by Group $I$, Group $II$ and Group $III$ are $\frac{1}{2},\frac{1}{4}$ and $\frac{1}{4}$ respectively. Randomly one of the messages is opened and found a message on sports. What is the probability that the message was from Group $III$.