A number X is chosen at random from the set of integers {−3, −2, −1, 0, 1, 2, 3}. What is the probability that the absolute value of X is less than 2, that is, that |X| is less than 2?

Introduction / Context:
This question focuses on a finite set of integers and uses the concept of absolute value. We choose an integer X from a small symmetric set and ask for the probability that its absolute value is less than 2. This tests understanding of absolute value inequalities and basic probability over a uniform discrete sample space.

Given Data / Assumptions:

• The set of possible values for X is {−3, −2, −1, 0, 1, 2, 3}.
• Each of the 7 integers is equally likely to be chosen.
• We need the probability that |X| less than 2.
• The absolute value of a number is its distance from zero on the number line.

Concept / Approach:
The inequality |X| less than 2 means that X lies strictly between −2 and 2. Thus the allowed integers are exactly those that have distance from zero less than 2 units. We first identify which integers in the given set satisfy this condition and then divide the number of such integers by the total number of integers in the set to get the probability.

Step-by-Step Solution:
Step 1: List the set of possible values of X explicitly: −3, −2, −1, 0, 1, 2 and 3.Step 2: Interpret |X| less than 2 as requiring that the distance from zero is less than 2, so X must lie strictly between −2 and 2.Step 3: The integers in the set that satisfy −2 less than X less than 2 are −1, 0 and 1.Step 4: Therefore, there are 3 favorable values of X: −1, 0 and 1.Step 5: The total number of possible values is 7, since the set has 7 elements.Step 6: The required probability is favorable outcomes divided by total outcomes = 3 / 7.
Verification / Alternative check:
We can check the complement event, where |X| is greater than or equal to 2. The integers with absolute value at least 2 are −3, −2, 2 and 3, which are 4 in number. Thus the probability of the complement event is 4 / 7. Adding this to the probability 3 / 7 for the original event gives 1, which confirms that our calculations account for all possible values of X correctly.

Why Other Options Are Wrong:
3/4 would require that 6 out of the 7 integers satisfy the inequality, which is not true. 4/5 would indicate 5.6 favorable outcomes, which makes no sense in this discrete context. 5/7 would correspond to 5 favorable integers, but only 3 actually satisfy |X| less than 2. 2/7 would only count two values, but we have identified three integers with the required property.

Common Pitfalls:
Students sometimes misread |X| less than 2 as |X| less than or equal to 2 and mistakenly include −2 and 2. Others misinterpret absolute value as a signed quantity rather than distance, leading to confusion. It is also easy to overlook 0 as a valid value when listing integers, but 0 clearly has absolute value 0, which is less than 2.

Final Answer:
The probability that the absolute value of X is less than 2 is 3/7.