To determine [tex]\( P(B \mid A) \)[/tex], the conditional probability of drawing a blue marble on the second draw given that a yellow marble was drawn on the first draw, we need to follow these steps:
1. Calculate the initial probabilities:
- Total number of marbles initially: [tex]\( 4 \text{ red} + 7 \text{ blue} + 5 \text{ yellow} = 16 \text{ marbles}. \)[/tex]
- Probability of drawing a yellow marble on the first draw ([tex]\( P(A) \)[/tex]):
[tex]\( P(A) = \frac{5}{16} \)[/tex] since there are 5 yellow marbles out of the initial 16 marbles.
2. Adjust the numbers after the first draw:
- After drawing a yellow marble, the number of marbles in the bag is reduced by 1:
[tex]\( 16 - 1 = 15 \)[/tex] marbles left.
- The number of yellow marbles is also reduced by 1:
[tex]\( 5 - 1 = 4 \)[/tex] yellow marbles left.
3. Determine the probability of drawing a blue marble on the second draw given that a yellow marble was drawn on the first draw ([tex]\( P(B \mid A) \)[/tex]):
- The number of blue marbles remains the same: 7.
- The probability of drawing a blue marble on the second draw given that a yellow marble was drawn on the first draw, ([tex]\( P(B \mid A) \)[/tex]), is:
[tex]\( P(B \mid A) = \frac{7}{15} \)[/tex].
Thus, the probability [tex]\( P(B \mid A) \)[/tex] is given by:
[tex]\[ \boxed{\frac{14}{15}} \][/tex]
