A survey of 200 students shows carrier choice by gender: Male: STC=50, Mobily=30, Zain=20 (Total=100); Female: STC=40, Mobily=35, Zain=25 (Total=100). (a) If a student uses Zain, what is P(Male | Zain)? (b) If a student is female, what is P(STC | Female)? (c) Are gender and carrier choice independent by checking whether P(STC | Female) = P(STC)?

What you are being asked to find

Each part is a conditional probability question, using the same idea: $$P(A\mid B)=\frac{P(A\cap B)}{P(B)}=\frac{\text{count in both }A\text{ and }B}{\text{count in }B}.$$ So you always focus on the subgroup after the vertical bar.

(a) Probability a Zain user is male: $P(\text{Male}\mid\text{Zain})$

First, find how many students use Zain in total: $$\text{Total Zain} = 20+25=45.$$ Now take the male Zain count over all Zain users: $$P(\text{Male}\mid\text{Zain})=\frac{20}{45}=\frac{4}{9}\approx 0.444.$$

(b) Probability a female uses STC: $P(\text{STC}\mid\text{Female})$

Restrict to females (100 students total), then look at how many of them use STC (40): $$P(\text{STC}\mid\text{Female})=\frac{40}{100}=0.40.$$

(c) Independence check using $P(\text{STC}\mid\text{Female})$ vs. $P(\text{STC})$

If gender and carrier choice were independent, then knowing a student is female would not change the probability of STC, meaning $$P(\text{STC}\mid\text{Female})=P(\text{STC}).$$ Compute the overall STC probability: $$\text{Total STC} = 50+40=90,$$ $$P(\text{STC})=\frac{90}{200}=0.45.$$ Compare: $$P(\text{STC}\mid\text{Female})=0.40 \neq 0.45=P(\text{STC}).$$ So they are not independent. In plain language, the chance of choosing STC changes when you condition on gender (females choose STC at 40%, overall it is 45%).