Probability of Product Not Greater Than 6 Given x and y from Set {1, 2, 3, 4}

In this article, we delve into the probability concept by analyzing the scenario where x and y, both taking values from the set {1, 2, 3, 4}, result in a product not greater than 6. We will use both mathematical reasoning and sample space representation to solve this problem.

Understanding the Problem

The problem states that x and y can take values from the set {1, 2, 3, 4}. We are interested in the probability that the product of x and y is not greater than 6. This involves analyzing the sample space and identifying the favorable outcomes.

A Fundamental Principle

The principle we follow is that the probability of an event occurring is the ratio of the number of favorable outcomes to the total number of possible outcomes. Mathematically, this is represented as:

Probability (P) Number of favorable outcomes / Total number of possible outcomes

Constructing the Sample Space

The sample space S includes all possible pairs (x, y) where x and y can take any value from the set {1, 2, 3, 4}. Therefore, the sample space S has a size of 16.

S {11, 12, 13, 14, 21, 22, 23, 24, 31, 32, 33, 34, 41, 42, 43, 44}

Identifying Favorable Outcomes

The event of interest is {11, 12, 13, 14, 21, 22, 23, 31, 32, 41}, which includes all pairs where the product of x and y is not greater than 6.

Let's verify these outcomes manually:

x y xy 1 1 1 1 2 2 1 3 3 1 4 4 2 1 2 2 2 4 2 3 6 3 1 3 3 2 6 4 1 4

From the table, we see that there are 10 pairs whose product is not greater than 6. Therefore, the probability is 10/16, which simplifies to 5/8.

Analyzing the Product Condition

Let's break down the logic further to understand the outcomes:

xy 1*1 1 xy 1*2 2 xy 1*3 3 xy 1*4 4 xy 2*1 2 (Repeated from the previous outcome) xy 2*2 4 xy 2*3 6 xy 3*1 3 xy 3*2 6 xy 4*1 4

The pairs (1, 2), (2, 1), and (3, 3) appear twice in the table due to the commutative property of multiplication. These ensure the total count of favorable outcomes remains 10.

Using the Permutation Formula

To quickly assess the number of permutations of the set {1, 2, 3, 4}, we can use the permutation formula:

Permutations n! / (n-r)! 4! / (4-2)! 4 * 3 * 2 * 1 / (2 * 1) 24 / 2 12

This accounts for the 12 unique permutations. However, since x and y can take the same values, we need to include the cases where x y, such as (1, 1), (2, 2), (3, 3), and (4, 4). This gives us an additional 4 cases, resulting in a total of 16 possible outcomes.

The final probability, therefore, remains 10/16, which simplifies to 5/8.

Conclusion

The probability that the product of x and y from the set {1, 2, 3, 4} is not greater than 6 is 5/8. Understanding the sample space and applying the correct principles of probability and permutations helps us arrive at the correct solution.