To solve for the conditional probability [tex]\( P(B \mid A) \)[/tex], we need to determine the probability of drawing an orange marble on the second draw given that a brown marble was drawn on the first draw.
Let's break this down step-by-step:
### Step 1: Determine the Total Number of Marbles
There are 5 orange marbles, 3 black marbles, and 6 brown marbles in the jar.
Total number of marbles:
[tex]\[ 5 + 3 + 6 = 14 \][/tex]
### Step 2: Probability of Drawing a Brown Marble on the First Draw (Event A)
The probability of drawing a brown marble on the first draw is:
[tex]\[ P(A) = \frac{\text{Number of brown marbles}}{\text{Total number of marbles}} = \frac{6}{14} \][/tex]
### Step 3: Update the Total Number of Remaining Marbles After Event A
Since one brown marble is removed after the first draw, the number of remaining marbles is:
[tex]\[ 14 - 1 = 13 \][/tex]
### Step 4: Probability of Drawing an Orange Marble on the Second Draw Given the First Was a Brown Marble (Event B Given A)
With a brown marble already removed, there remain 5 orange marbles and 13 total marbles. Thus, the probability of drawing an orange marble on the second draw given the first was a brown marble is:
[tex]\[ P(B \mid A) = \frac{\text{Number of orange marbles remaining}}{\text{Total number of remaining marbles}} = \frac{5}{13} \][/tex]
Thus, the conditional probability [tex]\( P(B \mid A) \)[/tex] is [tex]\( \frac{5}{13} \)[/tex].
Therefore, the correct answer is:
B) [tex]\(\frac{5}{13}\)[/tex]
