M/ 13 Module Practice Quiz
You are about to roll a six-sided die while at the same time randomly choose
from a bag of 4 blue marbles, 10 black marbles, and 7 orange marbles. What is
the probability of drawing a black marble and rolling a 1?
Submit your answer as an exact fraction or a decimal rounded to the hundredths
place.
Show your work here



Answer :

To find the probability of drawing a black marble and rolling a 1 on a six-sided die, we need to calculate the probability of each individual event happening and then find the combined probability of both events occurring together.

### Step-by-Step Solution:

1. Calculate the total number of marbles:

The bag contains:
- 4 blue marbles
- 10 black marbles
- 7 orange marbles

Therefore, the total number of marbles is:
[tex]\[ \text{Total marbles} = 4 + 10 + 7 = 21 \][/tex]

2. Calculate the probability of drawing a black marble:

There are 10 black marbles out of the total 21 marbles, so the probability ([tex]\(P(\text{Black})\)[/tex]) of drawing a black marble is:
[tex]\[ P(\text{Black}) = \frac{\text{Number of black marbles}}{\text{Total number of marbles}} = \frac{10}{21} \][/tex]

3. Calculate the probability of rolling a 1 on a six-sided die:

A standard six-sided die has 6 faces numbered from 1 to 6. The probability ([tex]\(P(\text{1})\)[/tex]) of rolling a 1 is:
[tex]\[ P(\text{1}) = \frac{1}{6} \][/tex]

4. Calculate the combined probability of both events:

Since drawing a marble and rolling a die are independent events, the combined probability ([tex]\(P(\text{Black and 1})\)[/tex]) is the product of the individual probabilities:
[tex]\[ P(\text{Black and 1}) = P(\text{Black}) \times P(\text{1}) = \left(\frac{10}{21}\right) \times \left(\frac{1}{6}\right) \][/tex]

Now, multiply the fractions:
[tex]\[ P(\text{Black and 1}) = \frac{10}{21} \times \frac{1}{6} = \frac{10 \times 1}{21 \times 6} = \frac{10}{126} \][/tex]

5. Simplify the fraction:

We can further simplify [tex]\(\frac{10}{126}\)[/tex] by finding the greatest common divisor (GCD) of 10 and 126, which is 2:
[tex]\[ \frac{10}{126} = \frac{10 \div 2}{126 \div 2} = \frac{5}{63} \][/tex]

So, the probability of drawing a black marble and rolling a 1 is [tex]\(\frac{5}{63}\)[/tex].

Thus, the final answer is:

[tex]\(\boxed{\frac{5}{63}}\)[/tex]