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]
### 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]
