To find the probability of parts having diameters larger than 138, given that the diameter is normally distributed with a mean ($\mu$) of 128 and a standard deviation ($\sigma$) of 0.77, we can use the standard normal distribution (z-score) to solve this.

Steps to Solve

1. Calculate the z-score:
   The z-score formula is: $z = \frac{X - \mu}{\sigma}$ where $X$ is the value we are interested in (138 in this case), $\mu$ is the mean, and $\sigma$ is the standard deviation.

   Plugging in the values: $z = \frac{138 - 128}{0.77} = \frac{10}{0.77} \approx 12.99$

2. Interpret the z-score:
   A z-score of 12.99 is extremely high, well above the range of typical z-scores (which usually range from about -3 to 3 in most practical applications of the normal distribution).

3. Find the probability:
   For a z-score this high, the probability of finding a part with a diameter greater than 138 is effectively 0. The standard normal distribution table or most statistical software would indicate a probability near zero for values so far from the mean.

Conclusion

The probability of finding parts with diameters larger than 138 is essentially 0.

Would you like further details on normal distributions or other related concepts?

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Here are five related questions to deepen your understanding:

1. What is the probability of finding parts with diameters smaller than 125?
2. How would we calculate probabilities for a range of diameters, such as between 126 and 130?
3. What proportion of parts would fall within one standard deviation of the mean?
4. How do standard deviation changes affect the spread in a normally distributed variable?
5. Can normal distributions have different shapes, or are they always bell-shaped?

Tip: The further a data point is from the mean, the smaller its probability in a normal distribution—especially beyond three standard deviations.