Parthasarathi Mukhopadhyay and Manabendra Nath Mukherjee Solutions for Exercise 2: EXERCISE

A survey revealed that $40$ boys out of $100$ and $165$ girls out of $1000$ secured $90\%$ or above marks in Mathematics in their annual examination. The survey was conducted on equal number of boys and girls from different places of a district. One of these boys and girls is chosen at random and it is found that the percentage of marks secured by the chosen one is less than $90$. Determine the probability that the chosen one is a girl.

Saheb tells truth $8$ times out of $10$ times. A die is tossed. Saheb reports that it is was $5$. What is the probability that it was actually $5 ?$

(Write answer in simplest fraction form)

A company has two plants to manufacture bicycle. The first plant manufactures $60\%$ of the bicycles and the second plant $40\%$. Out of that, $80\%$ of the bicycles are rated of standard quality at the first plant and $90\%$ of standard quality at the second plant. A bicycle is picked up at random and found to be of standard quality. Find the probability that it comes from the second plant.

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A test detects a disease correctly $80\%$ of time. But it fails to detect it correctly for $2\%$ of the time. A person is chosen at random from a large population of which $0.6\%$ have the disease. He underwent the test and was reported to have the disease. What is the probability that the person actually has the disease$?$

There are three coins. One is two headed coin, another is a biased coin that comes up heads $75\%$ of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed. If it shows head, what is the probability that it was the two headed coin$?$

(Write answer in simplest fraction form)

Let $d_1, d_2, d_3$ be three mutually exclusive diseases. Suppose a random sample of $15000$ patients contains $3000$ patients with disease $d_1$, $5000$ patients with disease $d_2$ and $7000$ patients with disease $d_3$. Also, $2500$ patients with disease $d_1$, $4000$ patients with disease $d_2$ and $2000$ patients with disease $d_3$ show a particular set of symptoms $\mathrm{S}$.  A patient is randomly chosen and it is found that he/she shows the symptom $\mathrm{S}$. On the basis of the information gathered, the doctor wishes to predict the disease which the patient is suffering from. What should be the doctor's legitimate prediction$?$

A picnic is arranged to be held on a particular day. The weather forecast says that there is $70\%$ chance of rain on that day. If it rains, the probability of a good picnic is $0.4$ and if it does not, the probability is $0.8$. What is the probability that the picnic will be good$?$

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A doctor is to visit a patient. From past experience, it is known that the probabilities that he will come by train, bus, scooter or by car are $\frac{3}{10}, \frac{1}{5}, \frac{1}{10}, \frac{2}{5}$ respectively. The probability that he will be late are $\frac{1}{4}, \frac{1}{3}, \frac{1}{12}$ if he comes by train, bus and scooter respectively; but if he comes by car he will not be late. When he arrives, he is late. What is the probability that he comes by train$?$

(Write answer in simplest fraction form)