If X, Y can take values from the set (1, 2, 3 ,4), find the probability that the product of X and Y is not greater than 6

1. Analyze the question

We are given two variables, $X$ and $Y$, which can each take any value from the set $\{1, 2, 3, 4\}$. We need to find the probability that their product ($X \times Y$) is not greater than 6 (i.e., $X \times Y \leq 6$).

2. Explain the concept

• Sample Space: Since $X$ has 4 choices and $Y$ has 4 choices, the total number of possible outcomes is $4 \times 4 = 16$.
• Probability Formula: $P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$
• Condition: We are looking for pairs $(X, Y)$ such that $X \times Y \leq 6$.

3. Show the solution

Let's list all possible outcomes $(X, Y)$ and identify those where the product is $\leq 6$:

| X \ Y | 1 | 2 | 3 | 4 | | :--- | :--- | :--- | :--- | :--- | | 1 | 1 (Yes) | 2 (Yes) | 3 (Yes) | 4 (Yes) | | 2 | 2 (Yes) | 4 (Yes) | 6 (Yes) | 8 (No) | | 3 | 3 (Yes) | 6 (Yes) | 9 (No) | 12 (No) | | 4 | 4 (Yes) | 8 (No) | 12 (No) | 16 (No) |

Counting the "Yes" outcomes:

• When $X=1$: Products are 1, 2, 3, 4 (All 4 are $\leq 6$)
• When $X=2$: Products are 2, 4, 6 (3 are $\leq 6$)
• When $X=3$: Products are 3, 6 (2 are $\leq 6$)
• When $X=4$: Product is 4 (1 is $\leq 6$)

Total favorable outcomes = $4 + 3 + 2 + 1 = 10$.

Calculating the probability: $P(XY \leq 6) = \frac{10}{16}$

Simplify the fraction by dividing both numerator and denominator by 2: $\frac{10 \div 2}{16 \div 2} = \frac{5}{8}$

4. Identify the answer

The calculated probability is $\frac{5}{8}$, which matches option A.

5. Tip

When dealing with probability questions involving two variables, a grid table is your best friend! It helps you visualize all possible outcomes and prevents you from missing any combinations. Always double-check if the question says "greater than" or "not greater than" (which includes the number itself) to avoid silly mistakes.