Suppose that a fair coin is tossed 500 times. Use the central limit theorem to approximate the probability of obtaining more than 300 heads

To approximate the probability of obtaining more than 300 heads in 500 tosses of a fair coin, we can apply the **Central Limit Theorem (CLT)**.

### Step-by-step solution:

#### 1. Define the problem:
Let \( X \) represent the number of heads in 500 tosses. Since each toss is independent and the coin is fair (probability of heads = 0.5), \( X \) follows a **binomial distribution**:
\[
X \sim \text{Binomial}(n = 500, p = 0.5)
\]

The mean \( \mu \) and variance \( \sigma^2 \) of a binomial distribution are given by:
- Mean: \( \mu = n \cdot p = 500 \cdot 0.5 = 250 \)
- Variance: \( \sigma^2 = n \cdot p \cdot (1 - p) = 500 \cdot 0.5 \cdot 0.5 = 125 \)
- Standard deviation: \( \sigma = \sqrt{125} \approx 11.18 \)

#### 2. Use the Central Limit Theorem:
The CLT tells us that for large \( n \), the binomial distribution \( X \) can be approximated by a normal distribution:
\[
X \sim N(\mu = 250, \sigma = 11.18)
\]

We are interested in finding the probability \( P(X > 300) \).

#### 3. Convert to the standard normal distribution:
We standardize the variable using the Z-score formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
For \( X = 300 \):
\[
Z = \frac{300 - 250}{11.18} = \frac{50}{11.18} \approx 4.47
\]

#### 4. Find the probability using the Z-table:
Now we need to find \( P(Z > 4.47) \). From standard normal tables or using a calculator, we know that the probability of getting a Z-score greater than 4.47 is extremely small:
\[
P(Z > 4.47) \approx 0.000004
\]

#### 5. Conclusion:
The probability of obtaining more than 300 heads in 500 tosses of a fair coin is approximately **0.000004**, or 0.0004%. This is a very small probability, indicating that getting more than 300 heads is highly unlikely.

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Would you like any more details on any of these steps or clarifications?

Here are 5 related questions to explore:

1. How does the Central Limit Theorem apply to other types of distributions?
2. What is the probability of getting exactly 300 heads?
3. How does increasing the number of tosses affect the accuracy of the normal approximation?
4. How would the result change if the coin was biased (e.g., \( p = 0.6 \))?
5. What is the probability of getting between 240 and 260 heads?

**Tip:** When using the normal approximation for binomial distributions, ensure that both \( np \) and \( n(1 - p) \) are greater than 5 to meet the approximation criteria well.

Suppose that a fair coin is tossed 500 times. Use the central limit theorem to approximate the probability of obtaining more than 300 heads.

Learn how to use the Central Limit Theorem to approximate the probability of getting more than 300 heads in 500 tosses of a fair coin. This detailed solution includes the use of binomial distribution, normal distribution, and Z-scores. Suitable for students in Grades 11-12.

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