Math Problem Statement

A number is selected at random from each of the sets {2, 3, 4} and {1, 3, 5}. Find the probability that the sum of the two numbers is greater than 3 and less than 7.

Solution

The problem in the image asks to find the probability that the sum of two randomly selected numbers from two sets is greater than 3 and less than 7.

The sets provided are:

  • Set A: {2, 3, 4}
  • Set B: {1, 3, 5}

We need to calculate the probability that the sum of two numbers, one from each set, lies between 3 and 7 (i.e., 3<sum<73 < \text{sum} < 7).

Step 1: List all possible outcomes

The possible sums from selecting one number from Set A and one number from Set B are:

&2 + 1 = 3 \\ &2 + 3 = 5 \\ &2 + 5 = 7 \\ &3 + 1 = 4 \\ &3 + 3 = 6 \\ &3 + 5 = 8 \\ &4 + 1 = 5 \\ &4 + 3 = 7 \\ &4 + 5 = 9 \\ \end{aligned}$$ So, the possible sums are: 3, 5, 7, 4, 6, 8, 5, 7, 9. ### Step 2: Identify favorable outcomes We are looking for sums greater than 3 and less than 7. These sums are 4, 5, and 6. From the list of sums, the favorable outcomes are: - 4 (from 3 + 1) - 5 (from 2 + 3, 4 + 1) - 6 (from 3 + 3) Thus, the favorable outcomes are: (3,1), (2,3), (4,1), and (3,3). So, there are 4 favorable outcomes. ### Step 3: Calculate the total possible outcomes Each number from Set A can be paired with any number from Set B. Since Set A has 3 elements and Set B has 3 elements, the total number of possible outcomes is: $$3 \times 3 = 9$$ ### Step 4: Probability The probability is the number of favorable outcomes divided by the total number of outcomes: $$\text{Probability} = \frac{4}{9}$$ Thus, the probability that the sum of the two numbers is greater than 3 and less than 7 is $$\frac{4}{9}$$. Would you like further details or have any questions? --- Here are 5 related questions: 1. What is the probability that the sum is exactly 6? 2. How would the probability change if Set A were {1, 2, 3}? 3. Can the result change if more numbers are added to Set B? 4. What is the complement of this probability? 5. What is the probability that the sum is greater than or equal to 7? **Tip:** When calculating probabilities, always start by determining all possible outcomes, then narrow down the favorable ones based on the problem's conditions.

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Math Problem Analysis

Mathematical Concepts

Probability
Set Theory
Addition

Formulas

Probability = Favorable outcomes / Total outcomes

Theorems

Basic probability principles

Suitable Grade Level

Grades 8-10