Ponderosa Paint and Glass carries three brands of paint. A customer wants to buy another gallon of paint to match paint she purchased at the store previously. She can't recall the brand name and does not wish to return home to find the old can of paint. So she selects two of the three brands of paint at random and buys them. Her husband also goes to the paint store and fails to remember what brand to buy. So he also purchases two of the three brands of paint at random. Determine the probability that both the woman and her husband fail to get the correct brand of paint. (Hint: Are the husband's selections independent of his wife's selections

Two events are independent of each other, this is called Compound Probability. Classical example is if two coins are filped, what is the probability of getting two heads in a row.?

Answer: each has 50% probability because there are two states only, either H or T. since we have two coins the probability of getting two heads in row is the product of obtaining heads for each individual coin (which is independent of each other). or is [tex]\frac{1}{2}[/tex]×[tex]\frac{1}{2}[/tex] = [tex]\frac{1}{4}[/tex] = 25%. (Notice!).

Back to our problem, now here we have 3 paint brands and one of them is the right paint brand and two of them are not correct brands. so the prabablity that the customer has picked not correct brand is [tex]\frac{2}{3}[/tex] and her husband has not picked correct brand is also [tex]\frac{2}{3}[/tex], since two probabilities are compound probabilities i.e indepandant of each other the probablity that both woman and her husband fail get correct brand is the product of the two and is = [tex]\frac{2}{3}[/tex]×[tex]\frac{2}{3}[/tex] = [tex]\frac{4}{9}[/tex].