I roll a pair of fair dice and compute the number of spots on the two sides facing up. Denote this total by X. The probability distribution of X is
X Prob
2 1/36
3 2/36
4 3/36
5 4/36
6 5/36
7 6/36
8 5/36
9 4/36
10 3/36
11 2/36
12 1/36
The probability that X is an even number is
An even number is any of 2,4,6,8,10,and 12 (within the sample space above).
So the probability of throwing a sum equal to an even number is the sum of the corresponding probabilities, namely:
1/36+3/36+5/36+...1/36.
To calculate the probability that X is an even number, we need to find the sum of the probabilities of getting 2, 4, 6, 8, 10, and 12.
P(X is even) = P(X = 2) + P(X = 4) + P(X = 6) + P(X = 8) + P(X = 10) + P(X = 12)
= 1/36 + 3/36 + 5/36 + 5/36 + 3/36 + 1/36
Finding the common denominator for the fractions, we have:
P(X is even) = (1 + 3 + 5 + 5 + 3 + 1)/36
= 18/36
Simplifying the fraction, we get:
P(X is even) = 1/2
Therefore, the probability that X is an even number is 1/2 or 50%.
To find the probability that X is an even number, we need to add up the probabilities of the even numbers in the distribution.
The even numbers in the distribution are 2, 4, 6, 8, 10, and 12.
Probabilities of even numbers:
P(X=2) = 1/36
P(X=4) = 3/36
P(X=6) = 5/36
P(X=8) = 5/36
P(X=10) = 3/36
P(X=12) = 1/36
To find the probability that X is an even number, we sum up these probabilities:
P(X is even) = P(X=2) + P(X=4) + P(X=6) + P(X=8) + P(X=10) + P(X=12)
= 1/36 + 3/36 + 5/36 + 5/36 + 3/36 + 1/36
= 18/36
= 1/2
Therefore, the probability that X is an even number is 1/2.