Maharashtra Board Solutions for Chapter: Probability, Exercise 6: MISCELLANEOUS EXERCISE - 9

For independent events $A$,$B$ and $C$, If $P(A\cap B)=\frac{1}{2}$, $P(B\cap C)=\frac{1}{3}$ and $P(C\cap A)=\frac{1}{6}$ then find $P\left(A\right)$, $P\left(B\right)$ and $P\left(C\right)$.

Let $A$ and $B$ be independent events with $P\left(A\right)=\frac{1}{4}$, and $P(A\cup B)=2P\left(B\right)-P\left(A\right)$. Find $P\left(B\right)$.

Let $A$ and $B$ be independent events with $P\left(A\right)=\frac{1}{4}$, and $P(A\cup B)=2P\left(B\right)-P\left(A\right)$. Find $P\left(\frac{B^{\text{'}}}{A}\right)$.

Find the probability that a year selected will have $53$ Wednesdays.

The chances of $P, Q$ and $R$ getting selected as principal of a college are $\frac{2}{5}, \frac{2}{5}, \frac{1}{5}$ respectively. Their chances of introducing $IT$ in the college are $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ respectively. Find the probability that $IT$ is introduced in the college after one of them is selected as a principal.

The chances of $P, Q$ and $R$ getting selected as principal of a college are $\frac{2}{5}, \frac{2}{5}, \frac{1}{5}$ respectively. Their chances of introducing $IT$ in the college are $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ respectively. Find the probability that $IT$ is introduced by $Q$.

Suppose that five good fuses and two defective ones have been mixed up. To find the defective fuses, we test them one by one, at random and without replacement. What is the probability that we are lucky and fine both of the defective fuses in the first two tests?

In a factory which manufactures bulbs, machines $A, B$ and $C$ manufacture respectively $25\%, 35\%$ and $40\%$ of the bulbs. Of their outputs, $5, 4$ and $2$ percent are respectively defective bulbs. A bulb is drawn at random from the product and is found to be defective. What is the probability that it is manufactured by the machine $B$?