Probability question with conditional probability concepts

1) Presume there is a measure designed to detect depression in adolescents. This measure detects depression in people who truly are depressed 75% of the time (hit rate), but it diagnoses depression in those who are not truly depressed 20% of the time (false positive). Presume there exists a population of adolescents (infinite in size) to whom this measure will be given. In this population, 10% are truly depressed and 90% are not. Sampled randomly.

1a)What is the probability that the first individual to whom measure is given is not truly depressed if not diagnosed as depressed.

1b) What is the probability the measure makes a correct diagnosis. (think about what constitute a correct diagnosis, so not detecting something when there's nothing there would count as well)

1c) what is the probability the measure makes a correct diagnosis in exactly 3 of the first five cases.

Solution Preview

1) Presume there is a measure designed to detect depression in adolescents. This measure detects depression in people who truly are depressed 75% of the time (hit rate), but it diagnoses depression in those who are not truly depressed 20% of the time (false positive). Presume there exists a population of adolescents (infinite in size) to whom this measure will be given. In this population, 10% are truly depressed and 90% are not. Sampled randomly.

Probability that a person is depressed = 0.10
Probability that not depressed = 1 - 0.10 = 0.90
Probability that a person is diagnosed as depressed ...

Solution Summary

The expert determines how to calculate conditional probabilities. Calculates of the binomial probability for k successes are provided.

2. For the following table, what is the value of :
a) P(A1)
b) P(B1│A2)
c) P(B2 and A3). Compute this as P(B2)*P(A3│ B2) . In what row and column will you find this answer? Rows are B1 & B2: columns are A1, A2 & A3.
Second Event
First Event
A1 A2 A3 Total
B1 2 1 3 6
B2 1 2 1 4

X~N(500,400) Determine the following
Random Variable X
a) P( X <= 515 )
b) P( X <= 515 | X > 450 ) (note: "|" implies given)
c) P( 20 < X^(1/2) <= 25 ) ( i.e. 20 < "square root of X" < 25 )
please clearly state each step for each part. The attached file states the problem again.

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Suppose an unrelated 77 year old man, 76 year old woman, and 82 year old woman are selected from a community.
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Show that if P(A|B) > P(A), then P(B|A) > P(B).
We know the following definitions:
ConditionalProbability:
The probability of event B given event A is P(B|A)=P(AandB)/P(A)
The probability of event A given event B is P(A|B)=P(Aand B)/P(B)
Independent Events:
T

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Please view the attachment for the rest of the question.

Suppose that x and y have the following joint probability distribution
X
2 4
_______________________________________________
1 0.10 0.15
Y 3 0.2 0.3
5 .1 .15
Find the marginal distribution of x a

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