How can we use the normal curve approximation to find the probability of getting between 185 and 210 heads when a coin is tossed 400 times, and what is the probability of getting exactly 205 heads?

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To solve the problem using the normal curve approximation, we first identify the parameters of our binomial distribution based on the coin-tossing scenario:

  • Number of trials (n): 400
  • Probability of success (p): 0.5 (since it’s a fair coin)

Next, we calculate the mean (μ) and standard deviation (σ) of the distribution:

  • Mean (μ): μ = n * p = 400 * 0.5 = 200
  • Standard Deviation (σ): σ = √(n * p * (1 – p)) = √(400 * 0.5 * 0.5) = √(100) = 10

a. Finding the Probability of Getting Between 185 and 210 Heads

To find the probability of obtaining between 185 and 210 heads (inclusive), we first convert these values to their corresponding z-scores:

  • Z-score for 185 heads: Z = (X – μ) / σ = (185 – 200) / 10 = -1.5
  • Z-score for 210 heads: Z = (X – μ) / σ = (210 – 200) / 10 = 1

Now, we look up these z-scores in the standard normal distribution table (or use a calculator):

  • P(Z < -1.5): Approximately 0.0668
  • P(Z < 1): Approximately 0.8413

To find the probability of getting between 185 and 210 heads, we subtract the two probabilities:

P(185 ≤ X ≤ 210) = P(Z < 1) - P(Z < -1.5) ≈ 0.8413 - 0.0668 = 0.7745

b. Finding the Probability of Getting Exactly 205 Heads

When calculating the probability of getting exactly 205 heads using the normal approximation, we can also use the continuity correction, adjusting our value slightly:

  • Z-score for 205 heads: Z = (204.5 – 200) / 10 = 0.45 (using 204.5 for continuity correction)

Now, we find the probability corresponding to this z-score:

  • P(Z < 0.45): Approximately 0.6736

To find P(X = 205), we can use the probability density function of the normal distribution:

P(X = 205) ≈ P(204.5 < X < 205.5)

Calculating the z-scores for 204.5 and 205.5:

  • Z-score for 204.5: (204.5 – 200) / 10 = 0.45
  • Z-score for 205.5: (205.5 – 200) / 10 = 0.55

Looking these up:

  • P(Z < 0.55): Approximately 0.7088

Now we can calculate the probability of getting exactly 205 heads:

P(204.5 < X < 205.5) = P(Z < 0.55) - P(Z < 0.45) ≈ 0.7088 - 0.6736 = 0.0352

In summary:

  • Probability of obtaining between 185 and 210 heads: 0.7745
  • Probability of obtaining exactly 205 heads: 0.0352

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