Question

Assume three cards are drawn from a standard 52-card deck without replacement. Answer each of the following questions

a) What is the probability that the third card will be the two of clubs?

b) Are your odds better for choosing the two of clubs on your first, second, or third draw?

c) How can you use this example to illustrate the difference between independent and dependent events?

d) How do the marginal, joint, and conditional probabilities change if we instead drew the cards with replacement?

Answer 1

Cards are drawn without replacement so that two of clubs couldn't have been drawn in earlier draws.

a) Probability that the third card drawn is two of clubs

(since 50 cards are left in deck after 2 draws)

b) Probability that the first card drawn is two of clubs

Probability that the second card drawn is two of clubs

b) Odd are better for choosing the two of clubs in third draw,

c) This happens because we are drawing without replacement, and thus size of deck keeps on decreasing. Hence the chances of getting two of club increases with each draw. So drawing without replacement is a dependent event. However if this had been drawing with replacement, each time deck will be full having 52 cards and thus probability of drawing two of clubs remains same in each draw which is and thus drawing with replacement is an independent event.

d) If instead cards are drawn with replacement, each draw is an independent event.

Hence joint probability = product of marginal probabilities = product of conditional probability.

Last equality arises from the fact that

for independent events.