Answer :
Certainly! Let's take this step by step.
### Step 1: Understand the Problem
We need to find the value of a binomial coefficient and a probability term given certain parameters.
The binomial coefficient we need to calculate is [tex]$\binom{6}{0}$[/tex], which represents the number of ways to choose 0 successes (k=0) out of 6 trials (n=6).
### Step 2: Calculate the Binomial Coefficient
The formula for the binomial coefficient is given by:
[tex]\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \][/tex]
Using our values [tex]\( n = 6 \)[/tex] and [tex]\( k = 0 \)[/tex]:
[tex]\[ \binom{6}{0} = \frac{6!}{0!(6-0)!} = \frac{6!}{0! \cdot 6!} = 1 \][/tex]
So, [tex]\(\binom{6}{0} = 1\)[/tex].
### Step 3: Calculate the Probability Term
The probability term we need to calculate is given by:
[tex]\[ p^k \cdot q^{n-k} \][/tex]
Where:
- [tex]\( p = 0.7 \)[/tex] (the probability of success)
- [tex]\( q = 0.3 \)[/tex] (the probability of failure, which is 1 - p)
- [tex]\( k = 0 \)[/tex]
- [tex]\( n = 6 \)[/tex]
Substituting the values:
[tex]\[ 0.7^0 \cdot 0.3^{6-0} = 0.7^0 \cdot 0.3^6 \][/tex]
Now, calculating the powers:
[tex]\[ 0.7^0 = 1 \][/tex]
[tex]\[ 0.3^6 \approx 0.000729 \][/tex]
So, the probability term is:
[tex]\[ 1 \cdot 0.000729 = 0.000729 \][/tex]
### Step 4: Combine the Results
Combining the binomial coefficient and the probability term, we get:
[tex]\[ \binom{6}{0} \cdot 0.7^0 \cdot 0.3^6 = 1 \cdot 0.000729 = 0.000729 \][/tex]
### Conclusion
So, the detailed step-by-step solution gives us:
[tex]\[ \binom{6}{0} = 1 \][/tex]
and
[tex]\[ 0.7^0 \cdot 0.3^{6-0} \approx 0.000729 \][/tex]
Thus, the results are [tex]\( (1, 0.0007289999999999998) \)[/tex].
### Step 1: Understand the Problem
We need to find the value of a binomial coefficient and a probability term given certain parameters.
The binomial coefficient we need to calculate is [tex]$\binom{6}{0}$[/tex], which represents the number of ways to choose 0 successes (k=0) out of 6 trials (n=6).
### Step 2: Calculate the Binomial Coefficient
The formula for the binomial coefficient is given by:
[tex]\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \][/tex]
Using our values [tex]\( n = 6 \)[/tex] and [tex]\( k = 0 \)[/tex]:
[tex]\[ \binom{6}{0} = \frac{6!}{0!(6-0)!} = \frac{6!}{0! \cdot 6!} = 1 \][/tex]
So, [tex]\(\binom{6}{0} = 1\)[/tex].
### Step 3: Calculate the Probability Term
The probability term we need to calculate is given by:
[tex]\[ p^k \cdot q^{n-k} \][/tex]
Where:
- [tex]\( p = 0.7 \)[/tex] (the probability of success)
- [tex]\( q = 0.3 \)[/tex] (the probability of failure, which is 1 - p)
- [tex]\( k = 0 \)[/tex]
- [tex]\( n = 6 \)[/tex]
Substituting the values:
[tex]\[ 0.7^0 \cdot 0.3^{6-0} = 0.7^0 \cdot 0.3^6 \][/tex]
Now, calculating the powers:
[tex]\[ 0.7^0 = 1 \][/tex]
[tex]\[ 0.3^6 \approx 0.000729 \][/tex]
So, the probability term is:
[tex]\[ 1 \cdot 0.000729 = 0.000729 \][/tex]
### Step 4: Combine the Results
Combining the binomial coefficient and the probability term, we get:
[tex]\[ \binom{6}{0} \cdot 0.7^0 \cdot 0.3^6 = 1 \cdot 0.000729 = 0.000729 \][/tex]
### Conclusion
So, the detailed step-by-step solution gives us:
[tex]\[ \binom{6}{0} = 1 \][/tex]
and
[tex]\[ 0.7^0 \cdot 0.3^{6-0} \approx 0.000729 \][/tex]
Thus, the results are [tex]\( (1, 0.0007289999999999998) \)[/tex].
