Question

Mr. Dupont is a professional wine taster. When given a French wine, he will identify it with probability $ 0.9 $ correctly as French and will mistake it for a Californian wine with probability $ 0.1 $ . When given a Californian wine, he will identify it with probability $ 0.8 $ correctly as Californian and will mistake it for a French wine with probability $ 0.2 $ . Suppose that Mr. Dupont is given ten unlabelled glasses of wine, three with French and seven with Californian wines. He randomly picks a glass, tries the wine and solemnly says: “French”. The probability that the wine he tasted was Californian, is nearly equal to,
A. $ 0.14 $
B. $ 0.24 $
C. $ 0.34 $
D. $ 0.44 $

Answer 1

Hint: The Californian probability is the ratio of product of probability of selecting California wine glass and probability of Dupont to say wrongly as French wine by the probability that Dupont say selected glass as French wine.

Complete step-by-step answer:
There are $ 7 $ Californian wine glasses and $ 3 $ French wine glasses.
The probability of selecting French wine glass $ P\left( {FG} \right) $ is $ \dfrac{3}{{10}} $ .
The probability of selecting California wine glass $ P\left( {CG} \right) $ is $ \dfrac{7}{{10}} $ when given the French wine.
The probability for selecting California wine glass $ P\left( F \right) $ is $ 0.9 $ when given French wine.
The probability of Dupont to say wrongly as Californian $ P\left( {{F_1}} \right) $ is $ 0.1 $ .
The probability of Dupont saying correctly as Californian wine $ P\left( F \right) $ is $ 0.8 $ .
The probability of Dupont to tell wrongly as French $ P\left( {{C_1}} \right) $ is $ 0.2 $ .
The probability that Dupont selected glass as French wine is equal to the sum probability for selecting the French as the French glass and say correctly as a French wine and probability of selecting Californian wine glass and wrongly said it as French wine.
 $ P\left( A \right) = P\left( {FG} \right) \times P\left( F \right) + P\left( {CG} \right) \times P\left( {{C_1}} \right) $
Substituting the values of all the probabilities we obtain,
 $ \begin{array}{c}
P\left( A \right) = \dfrac{3}{{10}} \times 0.9 + \dfrac{7}{{10}} \times 0.2\\
 = 0.041
\end{array} $
Hence, the probability of Dupont says selected glass for French of wine for california is equal to the ratio of product $ P\left( {CG} \right) $ and $ P\left( {{C_1}} \right) $ by $ P\left( A \right) $ .
\[P = \dfrac{{P\left( {CG} \right) \times P\left( {{C_1}} \right)}}{{P\left( A \right)}}\]
On substituting the values in the above expression, we get,
\[\begin{array}{l}
P = \dfrac{{P\left( {CG} \right) \times P\left( {{C_1}} \right)}}{{P\left( A \right)}}\\
P = \dfrac{{\left( {\dfrac{7}{{10}} \times 0.2} \right)}}{{0.41}}\\
P = 0.341
\end{array}\]
Therefore, the probability that the wine he tasted was Californian is 0.341.
So, the correct answer is “Option C”.

Note: Please be careful with the probability of Dupont say selected glass for French or wine for Californian is product probability of selecting California wine glass and probability of Dupont to tell wrongly as French by probability that Dupont selected glass as French wine.