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
$\mathrm{P}\left(\mathrm{X}=x\right)=\left\{\begin{array}{lll}kx& ,& \mathrm{if}x=0\mathrm{or}1\\ 2kx& ,& \mathrm{if}x=2\\ k\left(5-x\right)& ,& \mathrm{if}x=3\mathrm{or}4\\ 0& ,& \mathrm{if}x>4\end{array}\right.$
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