A manufacturing company of bulbs has 3 units $A, B$ and $C$ which produce $25 \%, 35 \%$ and $40 \%$ of the bulbs respectively. Out of the bulbs produced by $A, B, C$ units, $5 \%, 4 \%$ and $2 \%$ are defective, respectively. If a bulb is chosen at random and found to be defective, then the probability that it is produced by unit $B$ is

The probability distribution of a random variable $X$ is given below

$$ \begin{array}{ccccccc} \hline X & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline P\left(X=x_i\right) & \alpha & \alpha & \alpha & \beta & \beta & 0.3 \\ \hline \end{array} $$

If $\mu$ and $\sigma^2$ represent the mean and variance of $X$ and $\mu=4.2$, then $\sigma^2+\mu^2=$

The probability that a student gets distinction in a Mathematics test is $\frac{2}{3}$. If five such tests are conducted over a certain period of time, then the probability that he gets distinction in atleast 3 tests is

If $A$ and $B$ are events of a random experiment such that $P(A \cup B)=\frac{3}{4}, P(A \cap B)=\frac{1}{4}, P(\overline{\mathrm{~A}})=\frac{2}{3}$, then $P(\overline{\mathrm{~A}} \cap \mathrm{~B})=$