Question

Consider picking three cards (without replacement) from five cards marked with number 1 through 5, and observe the sequence.

(a) (2 pts) What is the total number of outcomes in the sample space?

(b) (3 pts) What is the conditional probability the second card is even given that the first card is even?

(c) (3 pts) What is the conditional probability the first two cards are even given the third card is odd?

(d) (2 pts) Consider the following two events: The first card is even and the third card is odd. Are these two events independent? Please explain.

Answer 1

a) since order is important number of outcomes =(5P3) =5!/(5-3)! =5*4*3=60

b) P(2nd card is even | first card is even) =1/4 (since if first card is even , there remains 1 even card in 4 remaining cards)

c)

P(3rd card is odd )=3/5

P(first 2 are even and 3rd is odd) =(2/5)*(1/4)*(3/3)=1/10

therefore P(first 2 odd | third card is odd) =(1/10)/(3/5) =1/6

d)

P(first card even) =2/5

P(third card odd) =3/5

P(first even and third odd) =P(1st even and 2nd even and third odd)+P(1st even and 2nd odd and third odd)=(2/5)*(1/4)*(3/3)+(2/5)*(3/4)*(2/3)=3/10

since P(first even and third odd) is not equal to P(first card even) *P(third card odd) , these events are not independent,