(a) Finds $k$ from the pmf's total-probability condition, then sums the pmf over the required ranges for each of the four probabilities; (b) fits a downward parabola with vertex at the fountain's peak through the origin, then evaluates it at the required horizontal distance. Both alternatives answered below.

(a) pmf of $X$ and the four probabilities

1. Find $k$. Total probability must equal $1$:

$k+2k+6k+5k+6k+10k=30k=1\ \Longrightarrow\ k=\dfrac1{30}$

2. Individual probabilities.

Check: $\dfrac{1+2+6+5+6+10}{30}=\dfrac{30}{30}=1$✓.

3. (i) $P(2<X<6)=P(X=3,4,5)$.

$f(3)+f(4)+f(5)=\dfrac{6+5+6}{30}=\dfrac{17}{30}$

4. (ii) $P(2\le X<5)=P(X=2,3,4)$.

$f(2)+f(3)+f(4)=\dfrac{2+6+5}{30}=\dfrac{13}{30}$

5. (iii) $P(X\le4)=P(X=1,2,3,4)$.

$f(1)+f(2)+f(3)+f(4)=\dfrac{1+2+6+5}{30}=\dfrac{14}{30}=\dfrac7{15}$

6. (iv) $P(3<X)=P(X=4,5,6)$.

$f(4)+f(5)+f(6)=\dfrac{5+6+10}{30}=\dfrac{21}{30}=\dfrac7{10}$

(b) Height of the fountain's water at $x=0.75$ m

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