+90 Today there is a 60% chance of rain, a 30% chance of lightning, and a 15% chance of lightning and rain together.
1) Determine if rain today and lightning today are independent events?
A)The two events are independent because P(lightning) ⋅ P(rain) = 0.15 does not equal P(lightning and rain) = 0.18.
B)The two events are not independent because P(lightning) ⋅ P(rain) = 0.15 does not equal P(lightning and rain) = 0.18.
C)The two events are independent because P(lightning) ⋅ P(rain) = 0.18 does not equal P(lightning and rain) = 0.15
D)The two events are not independent because P(lightning) ⋅ P(rain) = 0.18 does not equal P(lightning and rain) = 0.15
2) Now suppose that today there is a 60% chance of rain, a 30% chance of lightning, and a 10% chance of lightning if it’s raining. What is the chance of both rain and lightning today?
A)3%
B)6%
C)12%
D)18%
+6251 \(\text{If rain and lightning are independent then }\\ P[\text{rain AND lightning}]=P[\text{rain}]P[\text{lightning}]\\ 0.15 \neq 0.6 \cdot 0.3 = 0.18\\ \text{that's choice C}\)
\(P[\text{rain AND lightning}] = \\ P[\text{lightning |raining}]P[\text{raining}] = \\ (0.1)(0.6) = 0.06 = 6\%\\ \text{Choice B}\)
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