To determine the probability of drawing a red marble first and then a black marble without replacement, let's break down the problem step-by-step:
1. Total Marbles in the Bag:
- Number of black marbles [tex]\( = 6 \)[/tex]
- Number of red marbles [tex]\( = 7 \)[/tex]
Therefore, the total number of marbles is:
[tex]\[
6 + 7 = 13
\][/tex]
2. Probability of Drawing a Red Marble First:
- There are 7 red marbles out of the total 13 marbles.
[tex]\[
\text{Probability of drawing a red marble first} = \frac{7}{13}
\][/tex]
3. Probability of Drawing a Black Marble After a Red Marble:
- After drawing one red marble, there are now 12 marbles left in the bag.
- The number of black marbles remains 6 since we drew a red marble first.
[tex]\[
\text{Probability of drawing a black marble after a red marble} = \frac{6}{12} = \frac{1}{2}
\][/tex]
4. Combined Probability:
The combined probability of both events (drawing a red marble first and then a black marble) is found by multiplying the two individual probabilities:
[tex]\[
\text{Combined Probability} = \left(\frac{7}{13}\right) \times \left(\frac{1}{2}\right)
\][/tex]
[tex]\[
\text{Combined Probability} = \frac{7}{26}
\][/tex]
Thus, the probability of drawing a red marble first and then a black marble is [tex]\(\frac{7}{26}\)[/tex].
Given the choices:
(a) [tex]\(\frac{7}{26}\)[/tex]
(b) [tex]\(\frac{27}{26}\)[/tex]
(c) [tex]\(\frac{78}{84}\)[/tex]
(d) None
The correct answer is:
(a) [tex]\(\frac{7}{26}\)[/tex]
