Probability Q.

[nikvana]

Kurt, a painter, has 9 jars of paint:
4 are yellow
2 are red
rest are brown
Kurt will combine 3 jars of paint into a new container to make a new color, which he will name accordingly to the following conditions:

Brun Y if the paint contains 2 jars of brown paint and no yellow
Brun X if the paint contains 3 jars of brown paint
Jaune X if the paint contains at least 2 jars of yellow
Jaune Y if the paint contains exactly 1 jar of yellow

What is the probability that the new color will be Jaune

a) 5/42
b) 37/42
c) 1/21
d) 4/9
e) 5/9

Sol: 1. This has at least 2 yellow meaning..

a> there can be all three Y => 4c3
OR
b> 2 Y and 1 out of 2 R and 3 B => 4c2 x 5c1

Total 34

2.This has exactly 1 Y and remaining 2 out of 5 = > 4c1 x 5c2

Total 40
Total possibilities = (9!/3!6!) = 84
Adding the two probabilities: probability = 74/84 = 37/42


I solved it like
prob of it being Jaune X = 4/9 x 3/8 x 2/7 + 4/9 x 3/8 x 5/7 = 1/3
prob of it being Jaune Y = 4/9 x 5/8 x 4/7 = 10/63

Total prob of being jaune = 1/3 + 10/63 = 31/63.
Whats wrong with this method? :

[tanviet]

I do not understand the question. there is only 1 brown jar. How can we have 2 brown jar in the conditions.

[xyzabc123]

I did it differently.
if P (Jaune X/Y) needs to be calculated, all combinations containing at least one yellow jar need to be accounted for.
So the opposite will be calculating P(not Jaune X/Y). This corresponds to answering the question: what is the probability that a selection of three jars will have no yellow jar?

5/9 * 4/8 * 3/7 = 5/42

So P(Jaune X/Y) = 1 - 5/42 = 37/41

choose B

[mental]

I solved it like
prob of it being Jaune X = 4/9 x 3/8 x 2/7 + 4/9 x 3/8 x 5/7 = 1/3
prob of it being Jaune Y = 4/9 x 5/8 x 4/7 = 10/63

Total prob of being jaune = 1/3 + 10/63 = 31/63.

Sol II:
prob of it being Jaune X = (4/9 x 3/8 x 2/7) + (4/9 x 3/8 x 5/7 x 3C1) =
17/42
(3C1 = ways in which non yellow paint can be chosen: YYX, YXY, XYY)

prob of it being Jaune Y = (4/9 x 5/8 x 4/7 x 3C1) = 20/42
(3C1 = ways in which yellow paint can be chosen: xxY, xYx, Yxx)

17/42 + 20/42 = 37/42

..................................................................[/quote]

[mental]

I solved it like
prob of it being Jaune X = 4/9 x 3/8 x 2/7 + 4/9 x 3/8 x 5/7 = 1/3
prob of it being Jaune Y = 4/9 x 5/8 x 4/7 = 10/63

Total prob of being jaune = 1/3 + 10/63 = 31/63.

Sol II:
prob of it being Jaune X = (4/9 x 3/8 x 2/7) + (4/9 x 3/8 x 5/7 x 3C1) =
17/42
(3C1 = ways in which non yellow paint can be chosen: YYX, YXY, XYY)

prob of it being Jaune Y = (4/9 x 5/8 x 4/7 x 3C1) = 20/42
(3C1 = ways in which yellow paint can be chosen: xxY, xYx, Yxx)

17/42 + 20/42 = 37/42

..................................................................[/quote]