a pair of dice is rolled. is the event of rolling a sum of the dice is an odd number. Find the probability that the sum is a five or a ten, given that an odd sum is rolled.
The notation for conditional probability is P(B|A), read as the probability of B given A.
P(B|A) = P(A and B)/P(A)
for yours P(A) = P(odd sum is rolled)
P(B) = P(5 or 10)
ways to get 5: (1,4) (2,3) (3,2) (4,1) --> 4 ways
ways to get 10: (4,6) (5,5) (6,4) -- > 3 ways
P(A) = 18/36 = 1/2
P(A and B) = P((5or10) and odd) = 4/36
P(B|A) (4/36)/(1/2) = 2/9
To find the probability that the sum of two dice is an odd number, we need to consider the total number of outcomes where the sum is odd and divide it by the total number of possible outcomes.
Let's start by determining the total number of possible outcomes when rolling two dice. Each die has six sides, so there are 6 possible outcomes for each roll, resulting in a total of 6 x 6 = 36 possible outcomes when rolling two dice.
Next, let's find the number of outcomes where the sum is odd. For the sum of two dice to be odd, one die must show an even number, and the other die must show an odd number. There are 3 even numbers (2, 4, 6) and 3 odd numbers (1, 3, 5). To calculate the number of outcomes where the sum is odd, we multiply the number of even outcomes (3) by the number of odd outcomes (3), resulting in 3 x 3 = 9 possible outcomes.
Therefore, the probability of rolling an odd sum when rolling two dice is 9/36, which simplifies to 1/4.
Now let's consider the probability of rolling a five or a ten, given that an odd sum is rolled.
To find this probability, we need to determine the number of outcomes where the sum is either five or ten, given that the sum is odd. Since the sum is odd, we can conclude that the dice must show (2, 3) or (3, 2) to get a sum of five, or (4, 6) or (6, 4) to get a sum of ten.
From the previous calculations, we know that there are 9 outcomes where the sum is odd. Out of these 9 outcomes, two of them result in a sum of five (i.e., (2, 3) and (3, 2)), and one outcome results in a sum of ten (i.e., (6, 4)).
Therefore, the probability of rolling a sum of five or ten, given that an odd sum is rolled, is (2 + 1) / 9, which simplifies to 3/9 or 1/3.
So, the probability that the sum of the dice is a five or a ten, given that an odd sum is rolled, is 1/3.