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EXERCISE 11.3.1: Calculating conditional probabilities - two dice. A red and a blue die are thrown. Both dice are fair. The events A, B, and C are defined as follows: A: The sum on the two dice is even B: The sum on the two dice is at least 10 C: The red die comes up 5 (a) Calculate the probability of each individual event. That is, calculate p(A), p(B), and p(C). Solution For every outcome of the red die, there are three outcomes of the blue die that cause the sum to be even. (If the red die is even, the blue die must also be even. If the red die is odd, the blue die must also be odd.) Therefore |A| = 18. The size of the sample space is 36, and p(A) = 18/36 = 1/2. B = {(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)). Therefore |B| = 6, and p(B) = 6/36 = 1/6. There are six possible outcomes for the blue die, so |C| = 6, and p(C) = 1/6. (b) What is p(AIC)? Solution (c) What is p(BIC)? Solution (d) What is p(AIB)? Solution (e) Which pairs of events among A, B, and C are independent? Solution

          EXERCISE 11.3.1: Calculating conditional probabilities - two dice.
A red and a blue die are thrown. Both dice are fair. The events A, B, and C are defined as follows:
A: The sum on the two dice is even
B: The sum on the two dice is at least 10
C: The red die comes up 5
(a) Calculate the probability of each individual event. That is, calculate p(A), p(B), and p(C).
Solution
For every outcome of the red die, there are three outcomes of the blue die that cause the sum to be even. (If the red die is even,
the blue die must also be even. If the red die is odd, the blue die must also be odd.) Therefore |A| = 18. The size of the sample
space is 36, and p(A) = 18/36 = 1/2.
B = {(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)). Therefore |B| = 6, and p(B) = 6/36 = 1/6.
There are six possible outcomes for the blue die, so |C| = 6, and p(C) = 1/6.
(b) What is p(AIC)?
Solution
(c) What is p(BIC)?
Solution
(d) What is p(AIB)?
Solution
(e) Which pairs of events among A, B, and C are independent?
Solution

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EXERCISE 11.3.1: Calculating conditional probabilities - two dice.
A red and a blue die are thrown. Both dice are fair. The events A, B, and C are defined as follows:
A: The sum on the two dice is even
B: The sum on the two dice is at least 10
C: The red die comes up 5
(a) Calculate the probability of each individual event. That is, calculate p(A), p(B), and p(C).
Solution
For every outcome of the red die, there are three outcomes of the blue die that cause the sum to be even. (If the red die is even,
the blue die must also be even. If the red die is odd, the blue die must also be odd.) Therefore |A| = 18. The size of the sample
space is 36, and p(A) = 18/36 = 1/2.
B = (4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)). Therefore |B| = 6, and p(B) = 6/36 = 1/6.
There are six possible outcomes for the blue die, so |C| = 6, and p(C) = 1/6.
(b) What is p(AIC)?
Solution
(c) What is p(BIC)?
Solution
(d) What is p(AIB)?
Solution
(e) Which pairs of events among A, B, and C are independent?
Solution

Added by Amy C.

Computer Science and Information Technology

Trishna Knowledge Systems 2018 Edition

Instant Answer

Solved by Expert Kari Hasz

05/03/2022

Step 1

In other words, we need to find the probability that the sum on the two dice is even, given that the red die came up 5. Since the red die came up 5, there are three possible outcomes for the blue die that would result in an even sum: (5, 2), (5, 4), and (5, 6). Show more…

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Texts: EXERCISE 11.3.1: Calculating conditional probabilities - two dice A red and a blue die are thrown. Both dice are fair. The events A, B, and C are defined as follows: A: The sum on the two dice is even B: The sum on the two dice is at least 10 C: The red die comes up 5 (a) Calculate the probability of each individual event. That is, calculate p(A), p(B), and p(C). Solution: For every outcome of the red die, there are three outcomes of the blue die that cause the sum to be even. If the red die is even, the blue die must also be even. If the red die is odd, the blue die must also be odd. Therefore, |A| = 18. The size of the sample space is 36, and p(A) = 18/36 = 1/2. B = {4, 6, 5, 5, 6, 4, 6, 5, 6}. Therefore, |B| = 6, and p(B) = 6/36 = 1/6. There are six possible outcomes for the blue die, so |C| = 6, and p(C) = 1/6. (b) What is p(A|C)? Solution: (c) What is p(B|C)? Solution: (d) What is p(A|B)? Solution: (e) Which pairs of events among A, B, and C are independent? Solution:

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Transcript

00:01 A red and a blue dye are thrown.

00:03 Both dice are fair.

00:05 The events a, b, and c are defined as follows.

00:10 Event a, the sum of the numbers on the two dice is odd.

00:14 Event b, the sum of the numbers on the two dice is at most three.

00:18 And event c, the number on the blue dye, is a three.

00:24 Calculate the probability of each individual event.

00:26 So we'll start there.

00:28 So the probability that the sum is odd, i made myself a little addition grid because it's easier to see.

00:37 Now odd, half of them are going to be odd.

00:41 So that's a pretty easy probability to find.

00:45 One half, or we could use a decimal .5.

00:50 The probability that the sum is at most three.

00:56 So that means three or in this case two, and there's only three options here.

01:02 This probability is 3 out of 36, which would be 0 .083 repeating.

01:13 And the probability that the number on the blue dye is 3, we can see that in this column right here.

01:23 So all of these sums were made with a blue 3.

01:26 So there's 1, 2, 3, 4, 5, 6.

01:30 So 6 out of the 36, which would be .167.

01:39 Now we're going to calculate probability of a given b, the probability of b given c, and the probability of a given c...

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