Question

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 part a

Probability formula is \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
For part a, number of favorable outcomes (girls) \(n = 10\), total number of outcomes \(N=20\).
So \(P=\frac{10}{20}=\frac{1}{2}\)

Step2: Calculate probability for part b

If first contestant is a girl, then number of girls left \(n'=9\), total contestants left \(N' = 19\)
Using probability formula \(P=\frac{n'}{N'}\), we get \(P=\frac{9}{19}\)

Step3: Analyze independence for part c

Two events \(A\) (first is girl) and \(B\) (second is girl) are independent if \(P(B|A)=P(B)\)
\(P(B)=\frac{10}{20}=\frac{1}{2}\), \(P(B|A)=\frac{9}{19}\)
Since \(\frac{1}{2}
eq\frac{9}{19}\), the events are not independent

Answer:

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