Question

1) A deck of 72 cards contain either red or black cards. Suppose that it twice as likely to draw a black card than a red card. What is the probability of drawing 2 red cards and 2 black cards if you draw 4 cards with replacement?

2) A disease X has infected 1% of the total population. After a thorough testing of a large sample size, the accuracy of the test is given as follows: if a person is infected, it will alert ”positive” 99.5% of the time, and if a person is healthy, then it will alert ”negative” 99% of the time. If you took the test and it returned ”positive”, what is the probability that you are actually infected?

Answer 1

1) Drawing a black card is twice as likely to draw a red card. That means total number of black cards is $\frac{2}{3}(72)=48$

Total number of red cards is 24

the probability of drawing 2 red cards and 2 black cards if you draw 4 cards with replacement=

242 48 G -)2 = 0.0494 72 72

2)

0.01 0.99 with disease without disease 0.995 0.005 0.01 0.99 positive negative positive negative

If you took the test and it returned ”positive”, the probability that you are actually infected

= $P(with.disease \mid positive)=\frac{P(with.disease \cap positive)}{P(positive)}$

$P(with.disease \mid positive)=\frac{0.01*0.995}{0.01*0.995+0.99*0.01}=0.5013$