Final answer:
To calculate the probability, calculate the standard error of the mean and use z-scores to find the probabilities. The probability of the sample mean being within +/- 5 of the population mean is 0.6826, and the probability of it being within +/- 10 is 0.9544.
Step-by-step explanation:
To calculate the probability, we need to calculate the standard error of the mean first. The standard error of the mean is the standard deviation divided by the square root of the sample size. In this case, the standard error of the mean is 50 / sqrt(100) = 5. Then, we can calculate the z-scores for +/- 5 and +/- 10, which is the distance from the sample mean to the population mean in terms of standard errors. The z-score for +/- 5 is 5 / 5 = 1, and the z-score for +/- 10 is 10 / 5 = 2.
Using a standard normal distribution table or calculator, we can find the probability of getting a z-score less than -1 or greater than 1 for +/- 5, and the probability of getting a z-score less than -2 or greater than 2 for +/- 10. The probability for +/- 5 is 1 - 2 * (1 - 0.8413) = 0.6826, and the probability for +/- 10 is 1 - 2 * (1 - 0.9772) = 0.9544.