Question
You toss a coin 50 times and get 12 tails
Answers
GPT-4o mini
To analyze the results of tossing a coin 50 times and getting 12 tails, we can use the binomial distribution. In this scenario, we consider the following:
1. **Total trials (n)**: 50 (the number of times you tossed the coin)
2. **Number of successes (k)**: 12 (the number of tails you got)
3. **Probability of success (p)**: 0.5 (the probability of getting tails for a fair coin)
### Expected Outcomes
For a fair coin, the expected number of tails (successes) from 50 flips can be calculated as:
\[
E = n \cdot p = 50 \cdot 0.5 = 25
\]
### Variance and Standard Deviation
The variance (\(\sigma^2\)) of a binomial distribution can be calculated as:
\[
\sigma^2 = n \cdot p \cdot (1 - p)
\]
So,
\[
\sigma^2 = 50 \cdot 0.5 \cdot 0.5 = 12.5
\]
The standard deviation (\(\sigma\)) would then be:
\[
\sigma = \sqrt{12.5} \approx 3.54
\]
### Z-Score Calculation
The Z-score helps determine how unusual your result is:
\[
Z = \frac{(k - E)}{\sigma} = \frac{(12 - 25)}{3.54} \approx \frac{-13}{3.54} \approx -3.67
\]
### Conclusion
A Z-score of approximately -3.67 indicates that getting 12 tails in 50 tosses is quite unusual (far below the expected value). In fact, this Z-score corresponds to a very low probability for such an outcome if the coin is fair. In practical terms, it suggests a rare event under the assumption of a fair coin.
If you need further statistical analysis or context (such as calculating p-values or hypothesis testing), feel free to ask!
1. **Total trials (n)**: 50 (the number of times you tossed the coin)
2. **Number of successes (k)**: 12 (the number of tails you got)
3. **Probability of success (p)**: 0.5 (the probability of getting tails for a fair coin)
### Expected Outcomes
For a fair coin, the expected number of tails (successes) from 50 flips can be calculated as:
\[
E = n \cdot p = 50 \cdot 0.5 = 25
\]
### Variance and Standard Deviation
The variance (\(\sigma^2\)) of a binomial distribution can be calculated as:
\[
\sigma^2 = n \cdot p \cdot (1 - p)
\]
So,
\[
\sigma^2 = 50 \cdot 0.5 \cdot 0.5 = 12.5
\]
The standard deviation (\(\sigma\)) would then be:
\[
\sigma = \sqrt{12.5} \approx 3.54
\]
### Z-Score Calculation
The Z-score helps determine how unusual your result is:
\[
Z = \frac{(k - E)}{\sigma} = \frac{(12 - 25)}{3.54} \approx \frac{-13}{3.54} \approx -3.67
\]
### Conclusion
A Z-score of approximately -3.67 indicates that getting 12 tails in 50 tosses is quite unusual (far below the expected value). In fact, this Z-score corresponds to a very low probability for such an outcome if the coin is fair. In practical terms, it suggests a rare event under the assumption of a fair coin.
If you need further statistical analysis or context (such as calculating p-values or hypothesis testing), feel free to ask!