a pair of dice is rolled. find the probability that the sum of the dice is an odd number. also, find the probability that the sum is a 5 or 10 given that an odd sum is rolled
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To find the probability that the sum of a pair of dice is an odd number, we first need to determine the total number of possible outcomes. When rolling two dice, each die has six possible outcomes (numbers 1 through 6), so the total number of outcomes is 6 * 6 = 36.
Next, we need to list all the possible outcomes where the sum is an odd number. An odd number can only be obtained by adding an odd number and an even number or by adding two odd numbers. Here are the possible outcomes:
- Odd sum (Odd + Even): (1, 2), (1, 4), (1, 6), (2, 1), (2, 3), (2, 5), (3, 2), (3, 4), (3, 6), (4, 1), (4, 3), (4, 5), (5, 2), (5, 4), (5, 6), (6, 1), (6, 3), (6, 5) = 18 outcomes
Finally, divide the number of favorable outcomes (18) by the total number of possible outcomes (36) to find the probability:
Probability of obtaining an odd sum = 18/36 = 1/2
Now, let's find the probability of rolling a sum of either 5 or 10 given that an odd sum is rolled. We already know the total number of outcomes when rolling two dice is 36.
First, let's find the number of outcomes where the sum is 5:
- (1, 4), (4, 1) = 2 outcomes
Next, the number of outcomes where the sum is 10:
- (4, 6), (6, 4), (5, 5), (6, 6) = 4 outcomes
Now, let's find the number of outcomes where an odd sum is rolled. We already calculated that there are 18 outcomes.
Given that an odd sum is rolled, we need to find the probability of rolling a sum of 5 or 10. This is equivalent to finding the probability of rolling a sum of 5 or 10 among the 18 odd sum outcomes.
Number of favorable outcomes (sum of 5 or 10): 2 + 4 = 6
Now, divide the number of favorable outcomes (6) by the total number of outcomes (18) to find the probability:
Probability of obtaining a sum of 5 or 10, given an odd sum = 6/18 = 1/3