A company produces 1,000 refrigerators a week at three plants. Plant A produces 350 refrigerators a week, plant B produces 250 refrigerators a week, and plant C produces 400 refrigerators a week. Production records indicate that 5% of the refrigerators produced at plant A will be defective, 3% of those produced at plant B will be defective, and 7% of those produced at plant C will be defective. All the refrigerators are shipped to a central warehouse. If a refrigerator at the warehouse is found to be defective, what is the probability it was produced at plant A? Show your work if you would like full credit.
In a given country, records show that of the registered voters, 45% are Democrats, 35% are Republicans, and 20% are independents. In an election, 70% of the Democrats, 40% of the Republicans, and 80% of the independents voted in favor of a parks and recreation bond proposal. If a registered voter chosen at random is found to have voted in favor of the bond, what is the probability that the voter is a Republican? An independent? A Democrat? (Please show all work for full credit and have a legend for symbols).
1)
P(defective)=P(A)*P(defective|A)+P(B)*P(defective|B)+P(C)*P(defective|C) | |||
=0.35*0.05+0.25*0.03+0.4*0.07=0.053 |
P(A|defective)=P(A)*P(defective|A)/P(defective)=0.35*0.05/0.053 =0.3302 |
2)
P(defective)=P(Democrats)*P(defective|Democrats)+P(Republican)*P(defective|Republican)+P(Independent)*P(defective|Independent) | ||||||
=0.45*0.7+0.35*0.4+0.2*0.8=0.615 |
P(Democrats|defective)=P(Democrats)*P(defective|Democrats)/P(defective)=0.45*0.7/0.615=0.5122 |
P(Republican|defective)=P(Republican)*P(defective|Republican)/P(defective)=0.35*0.4/0.615=0.2276 |
P(Independent|defective)=P(Independent)*P(defective|Independent)/P(defective)=0.2*0.8/0.615=0.2602 |
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