Serve Speed Probability
1. **State the problem:** We want to find the probability that Novak Djokovic's first serve speed is at least 110 mph, given that the serve speed follows a normal distribution with mean $\mu = 115$ mph and standard deviation $\sigma = 6$ mph.
2. **Standardize the value:** Convert the serve speed 110 mph to a standard normal variable $Z$ using the formula:
$$Z = \frac{X - \mu}{\sigma} = \frac{110 - 115}{6} = \frac{-5}{6} = -0.8333$$
3. **Find the cumulative probability:** Using standard normal distribution tables or a calculator, find $P(Z \leq -0.8333)$. This is approximately 0.2023.
4. **Calculate the probability for at least 110 mph:** Since we want $P(X \geq 110)$, this is the complement of $P(X < 110)$:
$$P(X \geq 110) = 1 - P(Z \leq -0.8333) = 1 - 0.2023 = 0.7977$$
5. **Final answer:** The probability that Novak Djokovic's first serve speed is at least 110 mph is approximately **0.7977** (rounded to 4 decimal places).