a distribution where the mean is 90 and the standard eviation is 3 , find the largest fraction of the numbers that ould meet the following requirements. ess than 78 or more than 102 Of the numbers in the distribution, the fraction that is less than 78 or more than 102 is at most $\square$. (Type an integer or simplified fraction.)

Step 1 ：This is a problem of normal distribution. In a normal distribution, about 68% of values drawn from the distribution are within one standard deviation σ away from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. This is known as the 68-95-99.7 rule, or the empirical rule.

Step 2 ：Here, the mean is 90 and the standard deviation is 3. The numbers less than 78 or more than 102 are more than 4 standard deviations away from the mean. According to the empirical rule, the fraction of numbers that are more than 4 standard deviations away from the mean is very small, close to 0.

Step 3 ：We can calculate this fraction using the cumulative distribution function (CDF) of the normal distribution. The CDF gives the probability that a random variable drawn from the distribution is less than or equal to a certain value. We can use the CDF to find the probability that a number is less than 78 or more than 102.

Step 4 ：mean = 90, std_dev = 3, cdf_78 = 3.167124183311986e-05, cdf_102 = 3.167124183311998e-05, fraction = 6.334248366623985e-05

Step 5 ：The fraction of numbers that are less than 78 or more than 102 is approximately 0.00006334248366623985. This is a very small fraction, which is consistent with our expectation that the fraction of numbers more than 4 standard deviations away from the mean in a normal distribution is close to 0.

Step 6 ：Final Answer: The fraction of numbers that are less than 78 or more than 102 is approximately \( \boxed{0.00006334248366623985} \).