Probability given a mean and standard deviation of a random variable The number of transactions handled by a bank teller in a day is a random variable with a mean of 80 and standard deviation of 5. What can be said that the teller will handle at least 400 transactions in a day? What can be said about the probability that the teller will handle between 70 and 90 transactions in a day?

The probability that the teller will handle at least 400 transactions in a day is the following. $$P(|X| \ge 400) \le \frac{80}{400} = \frac{1}{5}$$

The probability that the teller will handle between 70 and 90 transactions in a day is the following. $$P(70 < X < 90) \ge 1-\frac{5^2}{100} = \frac{3}{4}$$