Question

Let, X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that

\[P\left( X = x \right) = \begin{cases}k\text{ x }  & , & \text{ if } x = 0 \text{ or }  1 \\ 2 \text{ kx }  & , & \text{ if }  x = 2 \\ k\left( 5 - x \right) & , & \text{ if } x = 3 \text{ or } 4 \\ 0 & , & \text{ if } x > 4\end{cases}\]

where k is a positive constant. Find the value of k. Also find the probability that you will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2 colleges.

Answer 1

The probability distribution of X is

X	0	1	2	3	4
P(X)	0	k	4k	2k	k

The given distribution is a probability distribution.

\[\therefore \sum_{} p_i = 1\]

⇒ 0 + k + 4k + 2k + k = 1
⇒ 8k = 1
⇒ k = 0.125

(i) P (getting admission in exactly one college) = P(X = 1) = k = 0.125

(ii) P (getting admission in at most 2 colleges) = P( X ≤ 2) = 0 + k + 4k = 5k = 0.625

(iii) P (getting admission in atleast 2 colleges) = P( X ≥ 2) = 4k + 2k + k = 7k = 0.875 .