P(E) = P((1, 4), (2, 3), (3, 2), (4, 1)) = 4/36 = 1/9
P(F) = 1/6
P(E
F) = P(sum is 5 and the red die shows 2) = P((3, 2)) = 1/36
A.
P(E) = P((1, 4), (2, 3), (3, 2), (4, 1)) = 4/36 = 1/9
P(F) = 1/6
P(EF) = P(sum is 5 and the red die shows 2) = P((3, 2)) = 1/36
Test for Independence
| P(EF) | = | P(E)P(F) | ? | ||
 36 | = | 9 | . | 6 |  |
Therefore, the events are not independent.
B.
P(E) = P((1, 4), (2, 3), (3, 2), (4, 1)) = 4/36 = 1/9
P(F) = 1/2
P(EF) = P(sum is 5 and the red die is even) = P((1, 4) , (3, 2)) = 2/36 = 1/18
Test for Independence
| P(EF) | = | P(E)P(F) | ? | ||
18 | = | 9 | . | 2 |  |
Therefore, the events are independent.
C.
P(E) = P((1, 4), (2, 3), (3, 2), (4, 1)) = 4/36 = 1/9
P(F) = P((1, 3), (2, 2), (3, 1)) = 3/36 = 1/12
P(EF) = P(sum is 5 and the sum is 4) = P(impossible event) = 0
Test for Independence
| P(EF) | = | P(E)P(F) | ? | ||
| = | 9 | . | 12 | |
|
Therefore, the events are not independent.
D.
P(E) = 1/2 (half the outcomes have an even sum)
P(F) = 1/2
P(EF) = P(sum is even and the red die is even) = 1/4 (there are 9 such outcomes)
Test for Independence
| P(EF) | = | P(E)P(F) | ? | ||
4 | = | 2 | . | 2 | |
Therefore, the events are independent.
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