Question

You have a standard deck of 52 cards and randomly draw two cards sequentially with replacement.

a) Compute the probability to get an ace in both trials.
b) Focusing on the first card only, compute the probability that the first card is either red (event A) or an ace (event B).
c) Are events A (the first card it red) and B (the first card is an ace) in the previous question independent?
Explain showing also the formulas you use.
d) Now imagine we select the two cards sequentially without replacement. Event A is the first card is red, and event B is the second card is an ace. Are the two events independent? Explain
why in words (not using formulas).
e)How would your answer to the previous point change if event A was the first card
is a 9" and event B was "the second card is an ace?

Answer 1

a. The probability of getting an ace in both trials when drawing two cards sequentially with replacement is 1/169. b. The probability that the first card is either red or an ace is 7/13. c. Events A (the first card is red) and B (the first card is an ace) are dependent. d. When selecting two cards sequentially without replacement, events A (the first card is red) and B (the second card is an ace) are dependent. e. The answer to the previous point remains the same even if event A is the first card is a 9 and event B is the second card is an ace.

Step-by-step explanation:

a. When drawing two cards sequentially with replacement, the probability of getting an ace in both trials can be calculated as follows:

Since there are four aces in a deck of 52 cards, the probability of drawing an ace in one trial is 4/52. Since we are replacing the card after each draw, the probability remains the same for the second trial:

P(Ace in first trial) * P(Ace in second trial) = (4/52) * (4/52) = 1/169

b. Focusing on the first card only, the probability that the first card is either red (event A) or an ace (event B) can be calculated as follows:

There are 26 red cards in a deck of 52 cards and 4 aces. Since there are 2 red aces, we need to subtract that to avoid double counting:

P(A) + P(B) - P(A and B) = (26/52) + (4/52) - (2/52) = 28/52 = 7/13

c. Events A (the first card is red) and B (the first card is an ace) are not independent. To determine independence, we need to compare the product of the probability of each event separately with the joint probability of both events occurring:

P(A) * P(B) = (26/52) * (4/52) = 1/52

P(A and B) = (2/52)

Since P(A) * P(B) is not equal to P(A and B), events A and B are dependent.

d. When selecting the two cards sequentially without replacement, event A (the first card is red) and event B (the second card is an ace) are not independent. The probability of the second card being an ace is influenced by the outcome of the first card draw. If the first card is an ace, the probability of the second card being an ace decreases as one ace has already been drawn.

e. If event A is the first card is a 9 and event B is the second card is an ace, the answer to the previous point remains the same. The dependence of the events is determined by the selection process without replacement, not the specific cards being chosen.