The question asks to arrange four intervals of a standard normal variate ($Z$) in increasing order of their associated probabilities. The standard normal distribution has a mean of 0 and a standard deviation of 1. The probability corresponds to the area under the standard normal curve.

Understanding Standard Normal Probabilities

The probability $P(a \le Z \le b)$ represents the area under the standard normal curve between the Z-scores $a$ and $b$. Key properties to remember are:

• The total area under the curve is 1.
• The curve is symmetric around the mean (Z=0).
• Areas (probabilities) decrease as the Z-scores move further away from the mean (0).

Analyzing the Intervals

Let's analyze the intervals provided:

• Interval A: $0 \le Z \le +1$. This is the area between the mean and 1 standard deviation above the mean.
• Interval B: $+1 \le Z \le +2$. This is the area between 1 and 2 standard deviations above the mean.
• Interval C: $+2 \le Z \le +3$. This is the area between 2 and 3 standard deviations above the mean.
• Interval D: $+3 \le Z < +\infty$. This is the area beyond 3 standard deviations above the mean.

Comparing Probabilities

Based on the properties of the standard normal distribution, the area under the curve decreases as we move further from the mean (Z=0). Therefore, we can compare the probabilities without exact calculation:

• Interval D ($+3 \le Z < +\infty$) is furthest from the mean, so it has the smallest probability.
• Interval C ($+2 \le Z \le +3$) is the next furthest.
• Interval B ($+1 \le Z \le +2$) is closer to the mean than C and D.
• Interval A ($0 \le Z \le +1$) is closest to the mean on the positive side, containing the largest area among these positive intervals.

Thus, the probabilities are ordered as follows:

This corresponds to the order D, C, B, A.

Ordering the Intervals

The intervals arranged in increasing order of probabilities are:

1. D: $+3 \le Z < +\infty$
2. C: $+2 \le Z \le +3$
3. B: $+1 \le Z \le +2$
4. A: $0 \le Z \le +1$

Therefore, the correct order is D, C, B, A.