An ellipse is inscribed in a circle and a point is chosen at random inside the circle. If the probability that this point lies outside the ellipse is $\frac{2}{3}$,then the eccentricity of the ellipse is $\frac{a\sqrt{b}}{c}$,where $\gcd(a, c) = 1$ and $b$ is a square-free integer. Find the value of $a \cdot b \cdot c$.

For the three events $A, B$ and $C$,$P$ (exactly one of the events $A$ or $B$ occurs) = $P$ (exactly one of the events $B$ or $C$ occurs) = $P$ (exactly one of the events $C$ or $A$ occurs) = $p$ and $P$ (all the three events occur simultaneously) = $p^2$,where $0 < p < 1/2$. Then the probability of at least one of the three events $A, B$ and $C$ occurring is

In a battery manufacturing factory, machines $P$, $Q$ and $R$ manufacture $20 \%$, $30 \%$ and $50 \%$ respectively of the total output. The chances that a defective battery is produced by these machines are $1 \%$, $1.5 \%$ and $2 \%$ respectively. If a battery is selected at random from the production, then the probability that it is defective is

Two coins are tossed. Let $A$ be the event that the first coin shows a head and $B$ be the event that the second coin shows a tail. The two events $A$ and $B$ are:

Three screws are drawn at random from a lot of $50$ screws containing $5$ defective ones. The probability of the event that all $3$ screws drawn are non-defective,assuming that the drawing is $(a)$ with replacement and $(b)$ without replacement,is respectively:

There are $3$ houses available in a residential area. $3$ people apply for the houses. If each person applies for a house without consulting others,what is the probability that all $3$ apply for the same house (in $/9$)?