To find the probability of a Major League Baseball game lasting between 210 and 220 minutes, we can use the properties of the normal distribution. Here's the process:

1. Convert the values to z-scores: The formula for converting a raw score $x$ to a z-score is: $z = \frac{x - \mu}{\sigma}$ where:

   • $\mu = 190$ (mean game duration)
   • $\sigma = 35$ (standard deviation)

   For 210 minutes: $z_1 = \frac{210 - 190}{35} = \frac{20}{35} \approx 0.57$

   For 220 minutes: $z_2 = \frac{220 - 190}{35} = \frac{30}{35} \approx 0.86$

2. Find the probabilities corresponding to these z-scores: Using standard normal distribution tables or a calculator:

   • The probability for $z_1 = 0.57$ is approximately 0.7157.
   • The probability for $z_2 = 0.86$ is approximately 0.8051.

3. Calculate the probability between 210 and 220 minutes: The probability is the difference between the two probabilities: $P(210 \leq X \leq 220) = P(z_2) - P(z_1) = 0.8051 - 0.7157 = 0.0894$

So, the probability of a game lasting between 210 and 220 minutes is approximately 0.09.

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Would you like further clarification, or have any questions?

Here are 5 related questions you might consider:

1. What is the probability of a game lasting more than 220 minutes?
2. How would the probability change if the mean game duration increased to 200 minutes?
3. What is the z-score for a game that lasts exactly 190 minutes?
4. How would you calculate the probability of a game lasting less than 210 minutes?
5. What is the area under the normal distribution curve for a game lasting exactly 190 minutes?

Tip: Always double-check your z-scores using a standard normal distribution table or calculator for precision.