Question

A computer is programmed to select two random numbers from 1 to 5. All possible outcomes are shown in the table. Each outcome has equal probability.



1 comma 1 1 comma 2 1 comma 3 1 comma 4 1 comma 5
2 comma 1 2 comma 2 2 comma 3 2 comma 4 2 comma 5
3 comma 1 3 comma 2 3 comma 3 3 comma 4 3 comma 5
4 comma 1 4 comma 2 4 comma 3 4 comma 4 4 comma 5
5 comma 1 5 comma 2 5 comma 3 5 comma 4 5 comma 5

Question
What is the probability that the first number selected was 3 or higher, given that the sum of the two numbers was 7?

Answer options with 4 options
A.
StartFraction 3 over 25 EndFraction

C.
StartFraction 1 over 5 EndFraction

B.
StartFraction 4 over 25 EndFraction

D.
StartFraction 3 over 4 EndFraction

Answer 1

To solve the problem, we can follow a systematic approach.

1. **Identify the total outcomes**: The table shows all possible pairs of numbers chosen from 1 to 5. There are \(5 \times 5 = 25\) potential outcomes since each selection is independent.

2. **Determine the outcomes where the sum of the two numbers is 7**: We’ll look for pairs \((x, y)\) such that \(x + y = 7\):
- \(2 + 5 = 7\) → (2, 5)
- \(3 + 4 = 7\) → (3, 4)
- \(4 + 3 = 7\) → (4, 3)
- \(5 + 2 = 7\) → (5, 2)

Therefore, the outcomes where the sum equals 7 are:
- (2, 5)
- (3, 4)
- (4, 3)
- (5, 2)

There are **4 outcomes** in which the sum is 7.

3. **Identify those outcomes where the first number is 3 or higher**: From the outcomes identified in the previous step, we need to find which ones have the first number (the first element of the pair) as 3 or higher:
- (3, 4) → First number = 3
- (4, 3) → First number = 4
- (5, 2) → First number = 5

Thus, the outcomes that meet this criterion are:
- (3, 4)
- (4, 3)
- (5, 2)

There are **3 outcomes** where the first number is 3 or higher.

4. **Calculating the probability**: We need the conditional probability of selecting an outcome where the first number is 3 or higher given that the sum of the numbers is 7. The probability \(P(A | B)\) can be calculated using the formula:
\[
P(A | B) = \frac{P(A \cap B)}{P(B)}
\]
Here, \(P(B)\) is the total number of outcomes where the sum is 7 (which is 4), and \(P(A \cap B)\) is the number of outcomes where the first number is 3 or higher and the sum is 7 (which is 3).

Therefore, the probability is:
\[
P(A | B) = \frac{3}{4}
\]

So, the answer is **D. \(\frac{3}{4}\)**.