As stated in the problem, we first need to check that the products n*p and n*q are each greater than or equal to 10. If this condition is not met, we are unable to proceed with the calculation.
If the condition is met, we can continue to calculate the mean and the standard deviation. The mean (μ) of a binomial distribution is simply n*p where n is the number of trials and p the probability of success.
For the standard deviation (σ), we can use the formula: sqrt(n * p * q). Here, n is the number of trials, p is the probability of success, and q is the probability of failure (which is 1-p).
So if we would have to calculated these values for specific values of n, p, q, and x, we would first check the previously mentioned condition. If the condition is met, we calculate the mean and the standard deviation.
Next, we adjust our probabilities. This is done because we are approximating a discrete probability with a continuous one. This adjustment is done in the following way:
- If we want to find the probability that X is equal to x (P(X=x)), we find the probability that X is less than or equal to x+0.5.
- If we want to find the probability that X is less than x (P(Xx)), we find the probability that X is greater than x+0.5.
To find these probabilities, we use the cumulative distribution function (CDF) of the normal distribution. This function gives the probability that a random variable X is less than or equal to a certain value. Using our mean and standard deviation, we can calculate these adjusted probabilities.
So in summary, first we ensure the condition of n*p and n*q each being greater or equal to 10 is met. Then we calculate the mean μ and standard deviation σ. And finally, we adjust our probabilities based on whether we are looking for the probability of X being equal, less or more than a given value.