The probability of getting between 115 and 137 heads when a coin is tossed 256 times is determined by converting the range to z-scores and utilizing the standard normal distribution to find the cumulative probability.
The question asks to use the normal curve approximation to find the probability of obtaining between 115 and 137 heads inclusive, when a coin is tossed 256 times. To solve this, we need to use the Normal Distribution as an approximation to the Binomial Distribution since the number of trials (n = 256) is quite large.
Let's find the mean (μ) and standard deviation (σ) for the number of heads. For a fair coin, the probability (p) of getting heads is 0.5. So the mean is μ = np = 256 * 0.5 = 128. The standard deviation is σ = √(np(1-p)) = √(256 * 0.5 * 0.5) ≈ 8.
To find the probability of obtaining between 115 and 137 heads inclusive, we convert the head counts to z-scores and look up the values associated with these z-scores in a standard normal distribution table or by using a calculator.
Z(115) = (115 - 128) / 8 = -1.625
Z(137) = (137 - 128) / 8 = 1.125
Using the standard normal distribution, we find the probabilities corresponding to these z-scores and then calculate the probability of landing between them. The result gives us the answer.
The probability of getting between 115 and 137 heads inclusive is found by looking up the z-scores of -1.625 and 1.125.