Let X denote the no. of hours a boy plays football on Sunday. Also, we know that :

$P (X=x)=\left\{\begin{matrix} k, & if & x=0\\k^2x, & if & x=1\, or \, 2\\k^2(x-1), & if & x=3\\k(x^2-13), & if & x=4 \end{matrix}\right. $

then which of the following is/are true ?

A. k =0.2

B. P (X=4)=0.6

C. P(X=2)=0.4

D. P(x ≥ 2)= 0.8

E. P(x= 5) = 0

Choose the correct answer from the options given below :

A, B and E only

The correct answer is Option (3) → A, B and E only

(A) $P(X=4)=k(4^2-13)=0.2(16-13)=0.2(3)=0.6$

(B) $P(X=2)=\frac{k}{2}(2)=\frac{0.2}{2}×2=0.2$

(E) $P(X=5)=0$ ⇒ Not defined by probability function

Sum of probabilities, $k+\frac{k}{2}(1)+\frac{k}{2}(2)+\frac{k}{2}(3-1)+k(4^2-13)=1$

$6.5k=1⇒k=\frac{1}{6.5}=0.2$