There is a deck of 12 cards numbered 1 through 12. One card is drawn at random from the
deck, look at it and then put it back in the deck. A total of two cards are drawn in this way.
Define the following events:
• A : get an even numbered card and an odd numbered card regardless of their order.
• B : The first card you draw is even numbered.
• C : The sum of the cards is even.
• D : The sum of the cards is 18.
a) Are events A and B independent? Justify your answer.
b) Are events A and C independent? Justify your answer.
c) Compute P(D).
d) Compute P(D|C).
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a) To determine if events A and B are independent, we need to check if the probability of A happening is affected by the occurrence of B.
Event A: Getting an even numbered card and an odd numbered card regardless of their order.
Event B: The first card you draw is even numbered.
To find P(A), we need to calculate the probability of getting one even numbered card and one odd numbered card.
Since there are 6 even numbered cards and 6 odd numbered cards in the deck, the probability of getting an even numbered card on the first draw is 6/12 = 1/2.
Now, we put the card back in the deck and draw again. The probability of getting an even numbered card on the second draw is also 1/2.
To find P(B), we only consider the first card drawn. Since there are 6 even numbered cards out of 12 total cards in the deck, the probability of getting an even numbered card on the first draw is 6/12 = 1/2.
To check if events A and B are independent, we calculate P(A|B), the probability of event A happening given that event B has already occurred. If P(A|B) is equal to P(A), then the events are independent.
P(A|B) = P(A intersection B) / P(B)
P(A intersection B) refers to the probability of getting an even numbered card and an odd numbered card on both draws. Since we can have two possible orders (odd-even or even-odd), the probability of A intersection B is 2 * (1/2 * 1/2) = 1/2.
P(B) = 1/2 (as calculated earlier)
Therefore, P(A|B) = (1/2) / (1/2) = 1, which is equal to P(A) = 1/2.
Since P(A|B) = P(A), events A and B are independent.
b) To determine if events A and C are independent, we need to check if the probability of A happening is affected by the occurrence of C.
Event A: Getting an even numbered card and an odd numbered card regardless of their order.
Event C: The sum of the cards is even.
To find P(A), we need to calculate the probability of getting one even numbered card and one odd numbered card.
As mentioned earlier, the probability of getting an even numbered card on the first draw is 1/2.
Now, for event C, we need to consider the different combinations of cards that result in an even sum.
Even + Even = Even
Even + Odd = Odd
Odd + Even = Odd
Odd + Odd = Even
Out of the four possible combinations, two result in an even sum (Even + Even and Odd + Odd).
Therefore, the probability of event C occurring is 2/4 = 1/2.
To check if events A and C are independent, we calculate P(A|C), the probability of event A happening given that event C has already occurred. If P(A|C) is equal to P(A), then the events are independent.
P(A|C) = P(A intersection C) / P(C)
P(A intersection C) refers to the probability of getting an even numbered card and an odd numbered card regardless of their order, given that the sum of the cards is even.
Since both even-odd and odd-even combinations result in an even sum, we have two possible outcomes out of four combinations.
Therefore, P(A intersection C) = 2/4 = 1/2.
P(C) = 1/2 (as calculated earlier)
Therefore, P(A|C) = (1/2) / (1/2) = 1, which is equal to P(A) = 1/2.
Since P(A|C) = P(A), events A and C are independent.
c) To compute P(D), we need to find the probability of getting a sum of 18 when drawing two cards.
To calculate this, we need to find the number of ways we can draw two cards that add up to 18 and divide it by the total number of possible outcomes.
The pairs that result in a sum of 18 are as follows:
(6, 12) and (12, 6)
Therefore, there are 2 possible outcomes that result in a sum of 18.
The total number of possible outcomes is the number of ways we can draw two cards out of twelve, which is given by the combination formula:
C(12, 2) = 12! / (2! * (12 - 2)!) = 66.
Therefore, P(D) = 2/66 = 1/33.
d) To compute P(D|C), we need to find the probability of getting a sum of 18 given that the sum of the cards is even.
Since the sum is even, there are three possible pairs that result in an even sum:
(6, 12), (12, 6), and (2, 16).
Out of these three pairs, only one pair results in a sum of 18: (6, 12).
Therefore, P(D|C) = 1/3.