Sure, let's break down the solution step-by-step for finding the probability of getting exactly 6 heads when a coin is flipped 12 times.
### Step 1: Total Sample Space, [tex]\( n(S) \)[/tex]
The total sample space [tex]\( n(S) \)[/tex] represents the number of all possible outcomes when flipping the coin 12 times. Since each flip has 2 possible outcomes (heads or tails), the total number of possible outcomes is:
[tex]\[ n(S) = 2^{12} \][/tex]
Given the result:
[tex]\[ n(S) = 4096 \][/tex]
### Step 2: Total Outcomes Wanted, [tex]\( n(A) \)[/tex]
The total outcomes wanted [tex]\( n(A) \)[/tex] is the number of ways to get exactly 6 heads in 12 flips. This is a combinatorial problem where we want to find in how many ways we can choose 6 positions out of 12 to be heads (the rest will be tails). This can be calculated using the binomial coefficient, which is:
[tex]\[ n(A) = \binom{12}{6} \][/tex]
Given the result:
[tex]\[ n(A) = 924 \][/tex]
### Step 3: Calculating the Probability, [tex]\( P(A) \)[/tex]
The probability [tex]\( P(A) \)[/tex] of getting exactly 6 heads in 12 flips is found using the formula:
[tex]\[ P(A) = \frac{n(A)}{n(S)} \][/tex]
Substituting the values we found:
[tex]\[ P(A) = \frac{924}{4096} \][/tex]
Given the result:
[tex]\[ P(A) = 0.2255859375 \][/tex]
### Conclusion
The total number of possible outcomes [tex]\( n(S) \)[/tex] when flipping a coin 12 times is 4096. The number of ways to get exactly 6 heads [tex]\( n(A) \)[/tex] is 924. Therefore, the probability of getting exactly 6 heads in 12 flips is:
[tex]\[ P(A) = 0.2255859375 \][/tex]
So the detailed steps lead to the following results:
- [tex]\( n(S) = 4096 \)[/tex]
- [tex]\( n(A) = 924 \)[/tex]
- [tex]\( P(A) = 0.2255859375 \)[/tex]
This is the probability of getting exactly 6 heads when a coin is flipped 12 times.
