Two six sided dice were rolled, and a friend tells us that the numbers that came up were identical. Find the probability that the sum of the numbers was as follows:
a) 4
b) 11
c) an even number
d) an odd number
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To find the probability of different sums, we need to first find the total number of outcomes and then determine the number of favorable outcomes.
Step 1: Find the total number of outcomes.
When rolling two six-sided dice, each die has 6 possible outcomes. Since we are rolling two dice simultaneously, the total number of outcomes is calculated by multiplying the number of outcomes for each die: 6 * 6 = 36
Step 2: Determine the number of favorable outcomes for each situation.
a) Find the probability of getting a sum of 4.
To get a sum of 4, the possible combinations are:
(1, 3), (2, 2), (3, 1) = 3 favorable outcomes
Thus, the probability of getting a sum of 4 is 3/36 = 1/12.
b) Find the probability of getting a sum of 11.
To get a sum of 11, the only combination is:
(5, 6) = 1 favorable outcome
Thus, the probability of getting a sum of 11 is 1/36.
c) Find the probability of getting an even number.
An even number can be obtained by either getting an even sum or an odd sum.
To get an even sum, we have the following favorable outcomes:
(1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (4, 6), (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6) = 18 outcomes
To get an odd sum, the remaining outcomes are odd:
(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
Thus, the probability of getting an even number is (18 + 18) / 36 = 36/36 = 1.
d) Find the probability of getting an odd number.
Since we already calculated the probability of getting an even number, the probability of getting an odd number can be found by subtracting the probability of getting an even number from 1:
Probability of getting an odd number = 1 - Probability of getting an even number = 1 - 1 = 0.
To summarize:
a) The probability of getting a sum of 4 is 1/12.
b) The probability of getting a sum of 11 is 1/36.
c) The probability of getting an even number is 1.
d) The probability of getting an odd number is 0.