Question

9 blue marbles and 5 green marbles are placed in a jar. two marbles are then drawn in succession without replacement. what is the probability that the first marble drawn is green, given that the second marble drawn is blue?
a) 0.7222
b) 0.5600
c) 0.3846
d) 0.6429
e) 0.4286
f) none of the above.

Explanation:

Step1: Identify total and colored marbles

Total marbles: \(9 + 5 = 14\). Blue: 9, Green: 5.

Step2: Use conditional probability formula

Let \(A\) = first marble green, \(B\) = second marble blue.
We need \(P(A|B)=\frac{P(A\cap B)}{P(B)}\).

• \(P(A\cap B)\): Green then Blue. Probability: \(\frac{5}{14}\times\frac{9}{13}=\frac{45}{182}\).
• \(P(B)\): Second marble blue. Two cases: Blue then Blue (\(\frac{9}{14}\times\frac{8}{13}\)) or Green then Blue (\(\frac{5}{14}\times\frac{9}{13}\)).

\(P(B)=\frac{9\times8 + 5\times9}{14\times13}=\frac{72 + 45}{182}=\frac{117}{182}\).

• \(P(A|B)=\frac{\frac{45}{182}}{\frac{117}{182}}=\frac{45}{117}=\frac{5}{13}\approx0.3846\).

Answer:

c) 0.3846