Roll four dice one time. What is the probability that the sum will be odd?Can anyone explain/show in detail how to find the answer to this problem?
I would guess 50% since it's always a 50% chance that you get an odd/even..
Can you explain in more detail or provide an example using all four dice?
(4 dice) for each dice 36 out comes
36 x 4 = 144
144/ 72 are odd = .5
I think that's right?
To find the probability that the sum of four dice rolled at once will be odd, you need to determine the total number of possible outcomes and the number of favorable outcomes.
Step 1: Determine the total number of possible outcomes
When rolling a single die, there are 6 possible outcomes – numbers 1, 2, 3, 4, 5, and 6. Since you're rolling four dice, you need to calculate the total number of outcomes by raising 6 to the power of 4, which is 6^4 = 1296.
Step 2: Determine the number of favorable outcomes
For the sum of four dice to be odd, one or three of the dice must show an odd number. There are four different possibilities for how many odd numbers will be obtained: 1 odd and 3 even, 3 odd and 1 even, 2 odd and 2 even, and 4 odd. Let's calculate the number of favorable outcomes for each possibility.
- 1 odd and 3 even:
To calculate the number of outcomes where exactly one die shows an odd number, we need to choose one die from the four to be odd, which can be done in 4 different ways. For the remaining three dice, they must show even numbers, which can be done in 3^3 = 27 ways. So the total number of outcomes where 1 odd and 3 even numbers are obtained is 4 * 27 = 108.
- 3 odd and 1 even:
To calculate the number of outcomes where exactly three dice show odd numbers, we need to choose three dice out of four, which can be done in (4 choose 3) = 4 ways. For each of these three dice, they can show an odd number in 3^3 = 27 ways. The remaining die must show an even number, which can happen in 3 ways. So the total number of outcomes where 3 odd and 1 even numbers are obtained is 4 * 27 * 3 = 324.
- 2 odd and 2 even:
To calculate the number of outcomes where exactly two dice show odd numbers, we need to choose two dice out of four, which can be done in (4 choose 2) = 6 ways. For these two dice, they can show odd numbers in 3^2 = 9 ways. The remaining two dice must show even numbers, which can happen in 3^2 = 9 ways. So the total number of outcomes where 2 odd and 2 even numbers are obtained is 6 * 9 * 9 = 486.
- 4 odd:
To calculate the number of outcomes where all four dice show odd numbers, we have just one possibility – all four dice showing an odd number, which can happen in 3^4 = 81 ways.
Step 3: Calculate the probability
Now that we have the number of favorable outcomes and the total number of possible outcomes, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
P(Sum of four dice is odd) = (Number of favorable outcomes) / (Total number of possible outcomes)
= (108 + 324 + 486 + 81) / 1296
= 999 / 1296
≈ 0.7715 or 77.15%
Therefore, the probability that the sum of four dice rolled at once will be odd is approximately 0.7715 or 77.15%.