The probability of an economic decline in the year 2020 is 0.23. There is a probability of 0.64 that we will elect a republican president in the year 2020. If we elect a republican president, there is a 0.35 probability of an economic decline. Let "D" represent the event of an economic decline, and "R" represent the event of election of a Republican president. Use this information to answer the following four questions: 1 Are R and D independent events?
2 What is the probability of electing a Republican president and an economic decline in 2020?
3 If we experience an economic decline in 2016, what is the probability that a Republican president will have been elected in 2020?
4 What is the probability of economic decline or a Republican president elected in 2020 or both?

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

42

step-by-step explanation:

256/6=42

you didn't post any answer choices, so its not 100% clear what your teacher is looking for

however, log(15/7) is equivalent to log(15) - log(7)

i'm using the rule log(x/y) = log(x) - log(y)