suppose a pair of dice is rolled. Consider the sums of numbers on the top of the dice. Find the probabilities:

a) P(5, given that the sum are odd)
b) P(odd, given that the sum rolled is 5)
c) P(7, given that one of the dice rolled was odd)

a) There are 36 possibilities. 18 of them are odd. Possible outcomes with a sum of 5 are: (1,4), (4,1), (2,3) and (3,2). The conitional probability is therefore 4/18 or 2/9.

b) If a sum of 5 is rolled, the sum is necessarily odd, so the conditional probability is 1.
c) The number of ways that only one die can be odd is:
(1,2), (1,4), (1,6), (3,2), (3,4), (3,6), (5,2), (5,4), (5,6), and the same combinations in reverse order, for a total of 18. Six of the combinations total 7, so the conditional probability is 6/18 = 1/3.

a) P(5, given that the sum are odd):

Well, because we know that the sum of the numbers on the dice is odd, we need to figure out which combinations of numbers on the dice would give us an odd sum of 5. Let's see... Ah, got it! The only combination that would work is a 2 on one dice and a 3 on the other. So, out of the 36 possible outcomes, there are two favorable outcomes.

Therefore, the probability of getting a sum of 5 given that the sum is odd is 2/36, which simplifies to 1/18.

b) P(odd, given that the sum rolled is 5):
Here, we know that the sum of the numbers on the dice is 5, and we want to find the probability of getting an odd number. In this case, we have three possible combinations: (1, 4), (3, 2), and (2, 3).

Out of the 36 possible outcomes, we have three favorable outcomes.

So, the probability of getting an odd number given that the sum is 5 is 3/36, which simplifies to 1/12.

c) P(7, given that one of the dice rolled was odd):
Alright, let me think about the possible outcomes here. If one of the dice rolled was odd, we can have the following combinations to get a sum of 7: (1, 6), (3, 4), (5, 2), and (2, 5).

Out of the 36 possible outcomes, we have four favorable outcomes.

So, the probability of getting a sum of 7 given that one of the dice rolled was odd is 4/36, which simplifies to 1/9.

Remember, these probabilities are just for fun, so don't go betting all your clown noses on them!

To find the probabilities, we can first determine the sample space, which represents all possible outcomes.

The sample space consists of all the possible sums that can be obtained from rolling a pair of dice. Since each die has 6 faces with numbers 1 to 6, there are 6 possible outcomes for each die, making a total of 6 x 6 = 36 possible outcomes.

a) P(5, given that the sum is odd):
To find this probability, we need to determine the subset of outcomes where the sum is odd, and within that subset, we find the probability of rolling a 5.

Subset of outcomes with odd sums: 2, 4, 6, 8, 10, 12
Subset of outcomes with odd sums where one of the dice shows a 5: 2, 4, 6, 8, 10, 12

There is only one outcome where the sum is 5 (one of the dice shows a 2 and the other shows a 3). Therefore, P(5, given that the sum is odd) = 1/6.

b) P(odd, given that the sum rolled is 5):
To find this probability, we need to determine the subset of outcomes where the sum is 5, and within that subset, we find the probability of rolling an odd number.

Subset of outcomes with a sum of 5: 2, 3; 3, 2; 1, 4; 4, 1
Subset of outcomes with a sum of 5 where at least one die shows an odd number: 1, 4; 4, 1

Out of the four possible outcomes where the sum is 5, two of them have at least one odd number. Therefore, P(odd, given that the sum rolled is 5) = 2/4 = 1/2.

c) P(7, given that one of the dice rolled was odd):
To find this probability, we need to determine the subset of outcomes where at least one die shows an odd number, and within that subset, we find the probability of rolling a sum of 7.

Subset of outcomes where at least one die is odd: 1, 1; 1, 2; 1, 3; 1, 4; 1, 5; 1, 6; 2, 1; 3, 1; 4, 1; 5, 1; 6, 1; 2, 3; 2, 5; 2, 7; 3, 2; 7, 2; 3, 4; 4, 3; 4, 5; 5, 4; 5, 6; 6, 5
Subset of outcomes where the sum is 7 and one die is odd: 1, 6; 2, 5; 3, 4; 4, 3; 5, 2; 6, 1

There are six possible outcomes where at least one die is odd, and out of those, six have a sum of 7. Therefore, P(7, given that one of the dice rolled was odd) = 6/6 = 1.

To find the probabilities, we need to determine the total number of possible outcomes and the number of favorable outcomes for each scenario.

In this case, rolling a pair of dice gives us a total of 36 possible outcomes since each die has 6 possible outcomes. To find the number of favorable outcomes in each scenario, we need to consider the specific conditions given.

a) P(5, given that the sum is odd):
To find the number of favorable outcomes for this scenario, we need to consider all the possible outcomes with an odd sum. These include (1, 4), (4, 1), (2, 3), (3, 2). There are four favorable outcomes. Therefore, the probability is calculated as:
P(5, given that the sum is odd) = favorable outcomes / total outcomes = 4 / 36 = 1 / 9.

b) P(odd, given that the sum rolled is 5):
To find the number of favorable outcomes for this scenario, we need to consider all the possible outcomes that result in a sum of 5. These include (1, 4), (4, 1), (2, 3), (3, 2). Out of these four outcomes, two of them have an odd number (1 and 3). Therefore, the probability is calculated as:
P(odd, given that the sum is 5) = favorable outcomes / total outcomes = 2 / 4 = 1 / 2.

c) P(7, given that one of the dice rolled was odd):
To find the number of favorable outcomes for this scenario, we need to consider all the possible outcomes where at least one die is odd. These include (1, 1), (1, 3), (1, 5), (1, 3), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5). Out of these ten outcomes, six of them have a sum of 7 (i.e., (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)). Therefore, the probability is calculated as:
P(7, given that one of the dice rolled was odd) = favorable outcomes / total outcomes = 6 / 10 = 3 / 5.

Remember, probability is calculated by dividing the number of favorable outcomes by the number of total outcomes.