Stats, probability I have the problem and the answers as well that the prof has given the class, i'm trying to work backwards to figure out how he solved it. Would appreciate if anyone could show me how he got the numbers (answers are highlighted) as all he gave was a sort explanation. 


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*Suppose that the probability of a worker receiving the minimum wage is 10%. Of
workers who receive the minimum wage, 50% are in low-income households. Of workers
who receive more than the minimum wage, 20% are in low-income households.


(a) What is the probability that a random household has low income?
Composing the 2x2 table of joint probabilities and adding up the entries gives 0.05
+ 0.18 = 0.23 or 23%
(b) If you are in a low-income household, what is the probability that you receive the
minimum wage?
The conditional probability is 0.05/.23 = 21.74%.
 A: Hi You can think like this -
Probability that a worker receiving min wage P(A) = 0.1
Probability that a worker receiving more than min wage P(B) = 0.9
Now probability that one household is low income provided the worker of that household receives min wages P(C/A) = 0.5 (C is the probability that the household is low income for minimum wages workers), again P(C/A) = P(A $\bigcap$ C) / P(A), so you can say P(A $\bigcap$ C) = 0.05 i.e total probability of the event : worker receives minimum wages "and" their household is low income = 5%
Similarly probability that one household is low income provided the worker of that household receives more than min wages P(D/B) = 0.2 (D is the probability that the household is low income for more than minimum wages workers), again P(D/B) = P(B $\bigcap$ D) / P(B), so you can say P(B $\bigcap$ D) = 0.18 i.e total probability of event : worker receives more than minimum wages "and" their household is low income = 18%
The event A:"worker receives minimum wages" and B:"worker receives more than minimum wages" are mutually exclusive, so you can come to conclusion any random house (not considering minimum wages) is low income household is : 0.18+0.05 = 0.23
And for second question "If you are in a low-income household, what is the probability that you receive the minimum wage?" A: you are in a low-income household
B: you receive the minimum wage
By conditional probability
P(B/A) = P(A $\bigcap$ B)/P(A)
Remember we previously derived "probability of worker receives minimum wages "and" their household is low income = 5%" & probability that you are in a low-income household = 23%
So P(B/A) = 0.05/0.23 = 21.74%
