A fair coin is tossed 400 times.Use the normal-curve approximation to find the probability of obtaining:
a.Between 185 and 210 heads inclusive.
b.Exactly 205 heads.
c.less than 176 or more than 227 heads.
To find the probabilities using the normal-curve approximation for a fair coin tossed 400 times, we can use the Central Limit Theorem. This theorem states that as the number of trials increases, the distribution of the sample mean approaches a normal distribution.
Let's denote "X" as the number of heads obtained in the coin tosses. For a fair coin, the probability of getting a head (success) is 0.5, and the probability of getting a tail (failure) is also 0.5. Thus, the mean (μ) and the standard deviation (σ) for X can be calculated as follows:
μ = n * p = 400 * 0.5 = 200
σ = sqrt(n * p * (1 - p)) = sqrt(400 * 0.5 * (1 - 0.5)) = sqrt(100) = 10
Now, we can proceed to find the probabilities using these mean and standard deviation values.
a. Between 185 and 210 heads inclusive:
To find this probability, we need to calculate the Z-scores for 185.5 and 210.5, and then find the difference between their respective cumulative probabilities in the standard normal distribution.
Z1 = (185.5 - μ) / σ = (185.5 - 200) / 10
Z2 = (210.5 - μ) / σ = (210.5 - 200) / 10
Using a standard normal table or a calculator, we can find the cumulative probabilities corresponding to these Z-scores and subtract them to get the desired probability.
b. Exactly 205 heads:
To find this probability, we need to calculate the Z-score for 205 and find its corresponding cumulative probability in the standard normal distribution.
Z = (205 - μ) / σ = (205 - 200) / 10
Using a standard normal table or a calculator, we can find the cumulative probability corresponding to this Z-score.
c. Less than 176 or more than 227 heads:
To find this probability, we need to calculate the Z-scores for 176.5 (lower end) and 227.5 (upper end), and then sum their respective cumulative probabilities in the standard normal distribution.
Z1 = (176.5 - μ) / σ = (176.5 - 200) / 10
Z2 = (227.5 - μ) / σ = (227.5 - 200) / 10
Using a standard normal table or a calculator, we can find the cumulative probabilities corresponding to these Z-scores and sum them to get the desired probability.
By following these steps and using either a standard normal table or a calculator, you can find the probabilities for each of the given scenarios.