Answer:
1) D and R are NOT independent events
2) The probability of electing a Republican president and an economic decline in 2020 is 0.224
3) If we experience an economic decline in 2016, the probability that a Republican president will have been elected in 2020 is 0.9739
4) the probability of economic decline or a Republican president elected in 2020 or both is 0.646
Step-by-step explanation:
Let "D" represent the event of an economic decline, and "R" represent the event of election of a Republican president
Given that;
P(D) = 0.23
P(R) = 0.64
Conditional P(D | R) = 0.35
1) Are R and D independent events?
we know that two events A & B are independent events if; P(B | A) = P(B)
here, P(D | R) = 0.35 and P(D) = 0.23
so; P(D | R) ≠ P(D)
Therefore D and R are NOT independent events
2) The probability of electing a Republican president and an economic decline in 2020;
we know that;
P(D | R) = P(D ∩ R) / P(R)
we substitute
0.35 = P(D ∩ R) / 0.64
P(D ∩ R) = 0.35 × 0.64
P(D ∩ R) = 0.224
Therefore, The probability of electing a Republican president and an economic decline in 2020 is 0.224
3) If we experience an economic decline in 2016, what is the probability that a Republican president will have been elected in 2020?
P(R | D) = P(D ∩ R) / P(D)
we substitute
P(R | D) = 0.224 / 0.23
P(R | D) = 0.9739
Therefore, If we experience an economic decline in 2016, the probability that a Republican president will have been elected in 2020 is 0.9739
4) the probability of economic decline or a Republican president elected in 2020 or both
P(D ∪ R) = P(D) + P(R) - P(D ∩ R)
we subtitute
P(D ∪ R) = 0.23 + 0.64 - 0.224
P(D ∪ R) = 0.646
Therefore, the probability of economic decline or a Republican president elected in 2020 or both is 0.646
