Question

Assume for the last 2 questions that there is an 18% chance of rain, a 6% chance of the daily low temperature reaching below 40 Fo, and a 23% of at least one of the 2 events happening.

7. Create either a probability table or tree of the data.

8. Show mathematically if these two events are either independent, positively dependent, or negatively dependent.

Answer 1

Q7) We are given here that:
P( rain) = 0.18 and P( low temp) = 0.06

P( rain or low temp) = 0.23

Using law of sum of probability, we have here:
P( rain and low temp.) = P(rain) + P(low temp) - P(rain or low temp)
P( rain and low temp) = 0.18 + 0.06 - 0.23 = 0.01

Therefore P(rain and no low temp.) = P(rain) - P(rain and low temp) = 0.18 - 0.01 = 0.17
P(no rain and low temp) = P(low temp.) - P(rain and low temp) = 0.06 - 0.01 = 0.05

Therefore the probability table here is obtained as:

	Rain	No Rain
Low Temperature	0.01	0.05
Not Low Temperature	0.17	1 - 0.01 - 0.05 - 0.17 = 0.77

Q8) We have here:
P(low temp. and low temp) = 0.01

P(low temp)P(rain) = 0.18*0.06 = 0.0108 which is not equal to P(low temp. and low temp)

Therefore the two variables are not independent here.