P(X < 8)=P((X - 8.1)/0.1 < (8 - 8.1)/0.1) = P(Z < 1)
That is, the probability that a chocolate bar falls below 8 oz is equal to the proportion of chocolate bars whose weights are *at most* +1 standard deviation above the mean.
Recall the empirical rule for normal distributions: approximately 68% of the distribution falls within 1 standard deviation of the mean, approximately 95% within 2 standard deviations, and approximately 99.7% within 3 standard deviations.
In other words,
P(-1 < Z < 1) = 0.68
P(-2 < Z < 2) = 0.95
P(-3 < Z < 3) = 0.997
which means
P(Z < -1 or Z > 1) = 1 - 0.68 = 0.32
P(Z < -2 or Z > 2) = 1 - 0.95 = 0.05
P(Z < -3 or Z > 3) = 1 - 0.997 = 0.003
The distribution is symmetric, which means
P(-1 < Z < 1) = 2 P(0 < Z < 1)
P(Z < 0) = P(Z > 0) = 0.5
The distribution is also continuous, which means
P(0 < Z < 1) = P(Z < 1) - P(Z < 0)
Putting all these facts together, we have
P(-1 < Z < 1) = 2 P (0 < Z < 1) = 2 (P(Z < 1) - P(Z < 0))
0.68 = 2 (P(Z < 1) - 0.5)
P(Z < 1) = 0.84
