Six cards-ace, king, queen of hearts; ace, king of diamonds; and king of clubs-are all placed face down a table. One card is to be selected at random. Find the probability:

a) of an ace, given that a diamond was turned
b) of an ace, given that a club was turned
c) of a heart, given that an ace was turned
d) of a club, given that a king was turned
e) Explain why it is true that P(A/C)=P(C/A)

After completing a through d, I thought I did them correctly unil I reached to e. Here are the answers I got: a) 2/5 b)2/5 c)1/5 d)1/5

Hummm, I do not agree with your answers a-d.

a) given that a diamond was turned, it must have been an ace or king. So for a) I get 1/2.
b) given that a club was turned, it must have been a king. So for b) I get a zero probability.

Now then, notationally, does P(A/C) mean probability of an Ace given a club was turned. If so, P(A/C)=0, and P(C/A)=0

how is that so when there are five cards faced down?

I get it. nevermind.

To find the probabilities, we can use conditional probability formulas. Let's go through each question step by step:

a) Probability of an ace, given that a diamond was turned:
To find this probability, we need to determine how many aces are there among the diamonds. In this case, there is only one ace among the diamonds. So, the probability of selecting an ace given that a diamond was turned is 1 out of the total number of diamonds, which is 1 out of 2. Therefore, the probability is 1/2.

b) Probability of an ace, given that a club was turned:
Similar to the previous case, there is only one ace among the clubs. So, the probability of selecting an ace given that a club was turned is 1 out of the total number of clubs, which is 1 out of 1. Therefore, the probability is 1.

c) Probability of a heart, given that an ace was turned:
Among the six cards, there are three hearts, and two of them are aces. So, the probability of selecting a heart given that an ace was turned is 2 out of the total number of aces, which is 2 out of 3. Therefore, the probability is 2/3.

d) Probability of a club, given that a king was turned:
Among the six cards, there is only one club, which is a king. So, the probability of selecting a club given that a king was turned is 1 out of the total number of kings, which is 1 out of 3. Therefore, the probability is 1/3.

Now, let's move on to e) and analyze why P(A/C) = P(C/A):

To express the probability P(A/C), we can interpret it as the probability of event A (e.g., selecting an ace) given that another event C (e.g., turning a club) occurred. Similarly, P(C/A) represents the probability of event C (e.g., turning a club) given that event A (e.g., selecting an ace) occurred.

In this case, both P(A/C) and P(C/A) have a denominator of the total number of cards of the respective selected event and a numerator with the number of cards that satisfy both the selected event and the condition.

If we consider the situation with equal probabilities, we can observe that the number of cards satisfying both events (ace and club) is the same as the number of cards satisfying both events (club and ace). Therefore, P(A/C) = P(C/A) is true.

Hence, P(A/C) and P(C/A) should be equal under the assumption of equal probabilities in the given scenario.