6. out of the 20 contestants in the state math championships, 10 are girls. for this round, each contestant…
6. 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.\na. what is the probability that the first contestant is a girl?\nb. if the first contestant is a girl, what is the probability that the second contestant is a girl?\nc. is the probability that the second contestant is a girl independent of the first contestant being a girl?
Answer
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