Question

****URGENT******

1A)  An event has four possible outcomes, A, B, C, and D. All of the outcomes are disjoint.

Given that P(B^c) = 0.2, P(A) = 0.1, and P(C) = 0.3, what is P(D)?

1B) A study was conducted on a potential association between drinking coffee and being diagnosed with clinical depression. All 18,832 subjects were female. The women were free of depression at the start of the study in 1996. Information was collected on coffee consumption and the incidence of clinical depression during the ten-year study period.

	≤ 1 cup coffee per week	2-6 cups coffee per week	TOTALS
Diagnosis of clinical depression	670	373	1043
No diagnosis of clinical depression	11,545	6244	17789
TOTALS	12,215	6,617	18,832

Are the following events independent?

Event LC: The event of drinking less than or equal to 1 cup of coffee per week

(Little Coffee = LC)

Event D: The event of a diagnosis of clinical depression

(Depression = D)

Round your calculations to four decimal places (or fewer) at each step.

There are multiple ways to test for independence. All involve the comparison of observed and expected probabilities based on probability theory.

In this context:

If the two probabilities are similar (identical to two decimal places), this is evidence of independence.

If the two probabilities are not similar (not identical to two decimal places), this is evidence of a lack of independence.

C) What do your results in (b) tell us, about the ways in which drinking very little coffee (0-1 cups per week) influences, or does not influence, the probability of depression for women in the study population?

Answer 1

Solution

Back-up Theory

If A1, A2, ........., Ak are mutually exclusive (disjoint) outcomes and these are the only outcomes which

make up an event, then

Σ(i = 1 to k)P(Ai) = 1. ........................................................................................................................................ (1)

If A and B are independent, P(A ∩ B) = P(A) x P(B) ..……………...................................................……………(2)

Now, to work out the solution,

Part 1(A)

Given,

‘An event has four possible outcomes, A, B, C, and D. All of the outcomes are disjoint.’, vide (1)

P(A) + P(B) + P(C) + P(D) = 1

Given that P(B^c) = 0.2, => P(B) = 0.8

Further given that: P(A) = 0.1, and P(C) = 0.3, P(D) = 1 – (0.1 + 0.8 + 0.3) = - 0.2, which is not acceptable since probability cannot be negative. As such, as given, the probability values are inconsistent.  Answer 1

Assuming a typo that it is P(B) and not P(B^c) = 0.2,

P(D) = 1 – (0.1 + 0.2 + 0.3) = 0.4, which is acceptable. Answer 1a

Part 1(B)

In the given table, if every entry is divided by the total frequency (i.e., 18832), we get the respective probability. Thus,

P(LC) = 12215/18832 = 0.649

P(D) = 1043/18832 = 0.055

P(LC ∩ D) = 670/18832 = 0.036

Hence, P(LC) x P(D) = 0.036

Since P(LC) x P(D) = P(LC ∩ D), vide (2), LC and D are independent. Answer 2

Part 1(C)

Since LC and D are independent, we conclude that drinking very little coffee (0-1 cups per week) does not influence, the probability of depression for women in the study population. Answer 3

DONE