What is the Probability of Flipping Heads Four Times in Twelve Coin Tosses?
Quick Answer
To calculate the probability of flipping heads exactly four times in twelve coin tosses, use the binomial probability formula, which gives you a result of approximately 0.1938 or 19.38%.
When dealing with a scenario where a fair coin is flipped multiple times, we often turn to the concept of binomial probability. This concept helps us understand the likelihood of a certain number of successes in a set number of trials. In this case, we want to find the probability of a fair coin landing heads up exactly four times when flipped twelve times.
### Step-by-Step Breakdown:
1. **Identify the Total Number of Trials (n)**: In this problem, the total number of flips, or trials, is 12. Therefore, n = 12.
2. **Determine the Number of Successes (k)**: We are looking for exactly 4 heads, which means k = 4.
3. **Calculate the Probability of Success (p)**: For a fair coin, the probability of getting heads in a single flip is 0.5.
4. **Calculate the Probability of Failure (q)**: Since we are dealing with a fair coin, the probability of getting tails is also 0.5.
5. **Use the Binomial Probability Formula**: The formula for binomial probability is:
$$P(X = k) = \binom{n}{k} \cdot p^k \cdot q^{n-k}$$
Where:
- \( \binom{n}{k} \) is the binomial coefficient, which represents the number of ways to choose k successes in n trials.
- p is the probability of success, and q is the probability of failure.
### Calculating the Values:
- First, calculate the binomial coefficient \( \binom{12}{4} \):
$$ \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 $$
- Next, calculate \( p^k \) and \( q^{n-k} \):
- \( p^k = (0.5)^4 = 0.0625 \)
- \( q^{n-k} = (0.5)^{12-4} = (0.5)^8 = 0.00390625 \)
- Now, plug these values into the binomial probability formula:
$$ P(X = 4) = 495 \cdot 0.0625 \cdot 0.00390625 $$
- This calculates to approximately 0.1938.
### Final Result:
Thus, the probability of flipping exactly four heads in twelve tosses of a fair coin is approximately 0.1938, or 19.38%.
### Real-World Applications:
Understanding binomial probabilities can help in various real-life situations, such as predicting outcomes in games, quality control in manufacturing, or even in genetics to determine the likelihood of certain traits appearing. This foundational knowledge in probability is crucial for making informed decisions based on statistical data.
If you would like to explore more about probabilities or related topics, feel free to ask!
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