Question

Suppose that for a car question, 1000 answers were submitted, of which 50 are correct.

a) Suppose also that the hosts randomly select two answers from those submitted with replacement. Calculate the probability that both selected answers are correct.

b) Suppose now that the hosts select the answers at random but without replacement. Use conditional probability to evaluate the probability that both answers selected are correct.

Answer 1

a)

$$P(C)=\frac{50}{1000}=0.05$$

$$P(C\ and \ C)=P(C)\cdot P(C)=P(C)^2=0.05^2=0.0025$$

b)

$$P(1C \ and \ 2C)=P(2C|1C)P(1C)$$

$$P(1C)=\frac{50}{1000}=0.05$$

$$P(2C|1C)=\frac{50-1}{1000-1}=\frac{49}{999}$$

Thus,

$$P(1C \ and \ 2C)=0.05\cdot \frac{49}{999}=0.00245$$