What is the probability that rolling four fair, six-sided dice produces an even sum?
even sum 2, 4, 6, 8 , 10, 12
ways 1,1 one way
3,1;2,2; 1,3 three ways
5,1;4,2;3,3;3,3;1,5;2;4 six ways
and continue...
So for the pr (2)=1/6*1/6
pr(4)=1/6*1/6*3
pr(6)=1/6*1/6*6
and continue
To have an even sum
1. all four must be even (E)
2. all four must be odd (O)
3 two must be even, two must be odd.
prob that one die lands even = 3/6 = 1/2
the same as for odd.
prob of EEEE = (1/2)^4
prob of OOOO = (1/2)^4
prob of EEOO = 4!/(2!2!) (1/2)^4
sum = 1/16 + 6(1/16) + 1/16 = 8/16 = 1/2
To find the probability of rolling an even sum with four fair, six-sided dice, we need to determine the number of favorable outcomes (even sums) and the total possible outcomes.
First, we know that each die has six possible outcomes (numbers 1 through 6) and that there are four dice. Therefore, the total number of possible outcomes is 6^4 (6 raised to the power of 4), which equals 1,296.
Next, we need to determine the number of favorable outcomes, which are the sums that result in an even value. An even sum can be obtained in three scenarios:
1. Even + Even + Even + Even
2. Odd + Odd + Odd + Even
3. Odd + Odd + Even + Even
For scenario 1, there are three even numbers (2, 4, and 6), and since we are using four dice, there would be 3^4 (3 raised to the power of 4) combinations, which results in 81 possibilities.
For scenario 2, there are three odd numbers (1, 3, and 5) and one even number (2 or 4 or 6) to choose from. Therefore, there are 3^3 (27) ways to choose the odd numbers and 3 ways to choose the even number, resulting in a total of 27 * 3 = 81 possibilities.
For scenario 3, there are two odd numbers (1 and 3) and two even numbers (2, 4, or 6) to choose from. Again, there are 2^2 (4) ways to choose the odd numbers and 3^2 (9) ways to choose the even numbers, resulting in 4 * 9 = 36 possibilities.
Adding up the favorable outcomes from all three scenarios gives us a total of 81 + 81 + 36 = 198 possibilities.
Therefore, the probability of rolling an even sum with four fair, six-sided dice is 198/1296, which simplified is 11/72 or approximately 0.1528 (rounded to four decimal places).
In summary, the probability is 11/72 or approximately 0.1528.