A person decides to toss a fair coin repeatedly until he gets a head. He will make at most $$3$$ tosses. Let the random variable $$Y$$ denotes the number of heads. The value of var $$\left\{ Y \right\},$$ where var $$\left\{ . \right\}$$ denotes the variance, equal

Let the probability density function of a random variable $$X,$$ be given as: $$${f_x}\left( x \right) = {3 \over 2}{e^{ - 3x}}u\left( x \right) + a{e^{4x}}u\left( { - x} \right)$$$
where $$u(x)$$ is the unit step function. Then the value of $$'a'$$ and Prob $$\left\{ {X \le 0} \right\},$$ respectively, are

Candidates were asked to come to an interview with $$3$$ pens each. Black, blue, green and red were the permitted pen colours that the candidate could bring. The probability that a candidate comes with all $$3$$ pens having the same colour is _______.

The probabilities that a student passes in Mathematics, Physics and Chemistry are $$m, p$$ and $$c$$ respectively. Of these subjects, the student has $$75$$% chance of passing in at least one, a $$50$$% chance of passing in at least two and a $$40$$% chance of passing in exactly two. Following relations are drawn in $$m, p, c:$$
$${\rm I}.$$ $$\,\,\,\,\,\,$$ $$p+m+c=27/20$$
$${\rm I}{\rm I}.$$ $$\,\,\,\,\,\,$$ $$p+m+c=13/20$$
$${\rm I}{\rm I}{\rm I}.$$ $$\,\,\,\,\,\,$$ $$\left( p \right) \times \left( m \right) \times \left( c \right) = 1/10$$