How to Calculate the Probability of Coin Flips: Heads Exactly Four Times
Quick Answer
To find the probability of getting heads exactly four times in twelve coin flips, use the binomial probability formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k). Here, n = 12, k = 4, and p = 0.5 for a fair coin.
Calculating the probability of a specific outcome in a series of coin flips can be an interesting exercise in understanding binomial probability. When you're trying to determine the probability of landing heads exactly four times in twelve flips, you're looking at a classic binomial probability problem.
The binomial probability formula is:
$$P(X = k) = inom{n}{k} p^k (1-p)^{n-k}$$
Where:
- **n** is the total number of trials (in this case, the number of coin flips, which is 12).
- **k** is the number of successful trials (the number of heads you want, which is 4).
- **p** is the probability of success in each trial (for a fair coin, this is 0.5, as the chances of landing heads or tails are equal).
### Step-by-Step Calculation
1. **Calculate the binomial coefficient \\(inom{12}{4}\\)**:
This represents the number of ways to choose 4 heads from 12 flips. The formula for the binomial coefficient is:
$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$
For our case:
$$\binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4!8!} = 495$$
2. **Calculate the probability of getting four heads**:
For the four heads, you calculate:
$$(0.5)^4 = 0.0625$$
3. **Calculate the probability of getting tails for the remaining flips**:
Since there will be 8 tails, you calculate:
$$(0.5)^{12-4} = (0.5)^8 = 0.00390625$$
### Putting It All Together
Now, combine these values to find the final probability:
$$P(X = 4) = \binom{12}{4} \times (0.5)^4 \times (0.5)^8$$
Substituting in the calculated values:
$$P(X = 4) = 495 \times 0.0625 \times 0.00390625$$
After performing the multiplication, you will find:
$$P(X = 4) \approx 0.192$$
### Real-World Application
Understanding binomial probability is crucial not just in mathematics but also in fields like statistics, finance, and any area where you need to predict outcomes based on a set of independent events. This knowledge can help in making informed decisions based on probabilities, such as in risk assessment or game theory.
### Conclusion
In summary, by breaking down the problem into manageable steps and applying the binomial probability formula, you can accurately calculate the likelihood of achieving a specific outcome in a series of independent trials. This method can be applied to various scenarios beyond coin flips, making it a valuable tool in your mathematical toolkit.
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