Given $SetA=\left\{2,3,4,5\right\}$ and Set $B=\left\{11,12,13,14,15\right\}$, two numbers are randomly selected, one from each set. What is the probability that the sum of the two numbers equals 16?

This problem involves calculating the probability of a specific event occurring when selecting numbers from two different sets. Probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes.

Understanding the Sets and Outcomes

We are given two sets:

- Set A contains the numbers: $\left\{2,3,4,5\right\}$. The total count of numbers in Set A is 4.
- Set B contains the numbers: $\left\{11,12,13,14,15\right\}$. The total count of numbers in Set B is 5.

We need to select one number randomly from Set A and one number randomly from Set B. The total number of possible pairs we can form is the product of the number of elements in each set.

Total possible outcomes = (Number of elements in Set A) $\times$ (Number of elements in Set B)

Total possible outcomes = $4\times 5=20$.

Identifying Favorable Outcomes

We are looking for the probability that the sum of the two selected numbers equals 16. Let's find the pairs of numbers (one from Set A, one from Set B) that add up to 16:

- If we select 2 from Set A, we need $16-2=14$ from Set B. The number 14 is present in Set B. So, (2, 14) is a favorable outcome.
- If we select 3 from Set A, we need $16-3=13$ from Set B. The number 13 is present in Set B. So, (3, 13) is a favorable outcome.
- If we select 4 from Set A, we need $16-4=12$ from Set B. The number 12 is present in Set B. So, (4, 12) is a favorable outcome.
- If we select 5 from Set A, we need $16-5=11$ from Set B. The number 11 is present in Set B. So, (5, 11) is a favorable outcome.

The favorable outcomes are the pairs: (2, 14), (3, 13), (4, 12), and (5, 11).

The number of favorable outcomes is 4.

Calculating the Probability

The probability of an event is calculated using the formula:

$$P(\text{Event})=\frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}$$

In this case, the event is that the sum of the two selected numbers equals 16.

Number of Favorable Outcomes = 4

Total Number of Possible Outcomes = 20

$$P(\text{Sum = 16})=\frac{4}{20}$$

Simplifying the fraction:

$$P(\text{Sum = 16})=\frac{1}{5}$$

Converting the fraction to a decimal:

$$P(\text{Sum = 16})=0.20$$