The probability of obtaining heads on a biased coin is 0.4. The coin is tossed 600 times.
a. i. Write down the mean number of heads (240)
ii. Find the standard deviation of the number of heads (12)
b. Find the probability that the number of heads obtained is less than one standard deviation away from the mean.
Part b is what I need help with. I have identified that the range of successes is between 228 and 252, and I know that I'm supposed to use the binomial distribution function, but we didn't learn in class what that is or how to use it.
To find the probability that the number of heads obtained is less than one standard deviation away from the mean, we can use the normal approximation to the binomial distribution. This approach assumes that the distribution of the number of heads follows a normal distribution.
1. Calculate the mean (µ) and standard deviation (σ) of the number of heads:
Given that the probability of obtaining heads is 0.4 and the coin is tossed 600 times:
µ = n * p = 600 * 0.4 = 240
σ = sqrt(n * p * (1 - p)) = sqrt(600 * 0.4 * 0.6) = 12
2. Define the range of successes within one standard deviation away from the mean:
Lower bound = µ - σ = 240 - 12 = 228
Upper bound = µ + σ = 240 + 12 = 252
3. Calculate the cumulative probability using the normal distribution function:
In this case, we want to find the probability of obtaining fewer than 252 heads but more than 228 heads.
P(X < 252) - P(X < 228)
We can use a standard normal distribution table or a statistical software to find these probabilities. However, since you mentioned not being introduced to the binomial distribution function, here's an alternative method:
Standardize the upper and lower bounds of the range:
Z1 = (X1 - µ) / σ = (228 - 240) / 12 = -1
Z2 = (X2 - µ) / σ = (252 - 240) / 12 = +1
Using the standard normal distribution table, find the probabilities corresponding to these Z-scores:
P(Z < -1) = 0.1587
P(Z < 1) = 0.8413
Finally, subtract the lower probability from the upper probability to find the desired probability:
P(X < 252) - P(X < 228) = 0.8413 - 0.1587 = 0.6826
Therefore, the probability that the number of heads obtained is less than one standard deviation away from the mean is approximately 0.6826 or 68.26%.