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

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 _______.

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

Given Set $$\,\,\,A = \left\{ {2,3,4,5} \right\}\,\,\,$$ and Set $$\,\,\,B = \left\{ {11,12,13,14,15} \right\},\,\,\,$$ two numbers are randomly selected, one from each set. What is the probability that the sum of the two numbers equal $$16?$$