Question

1. out of the 20 contestants in the state math championships, 10 are girls. for this round, each contestant gets asked one question. the first question goes to a randomly chosen contestant.

a. what is the probability that the first contestant is a girl?
b. if the first contestant is a girl, what is the probability that the second contestant is a girl?
c. is the probability that the second contestant is a girl independent of the first contestant being a girl?

Explanation:

Step1: Calculate probability for first contestant

Probability is number of favorable outcomes over total outcomes. Total contestants are 20 and girls are 10. So probability $P_1=\frac{10}{20}=\frac{1}{2}$.

Step2: Calculate probability for second contestant given first is girl

If first contestant is a girl, then 9 girls are left and 19 contestants are left. So probability $P_2 = \frac{9}{19}$.

Step3: Check for independence

Two events are independent if $P(A\cap B)=P(A)\times P(B)$. Here, the probability that the second contestant is a girl changes depending on whether the first contestant is a girl. So they are not independent.

Answer:

a. $\frac{1}{2}$
b. $\frac{9}{19}$
c. No