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.
what is the probability that the first number selected was 3 or higher, given that the sum of the two numbers was 7?
a. $\frac{3}{25}$
b. $\frac{4}{25}$
c. $\frac{1}{5}$
d. $\frac{3}{4}$

Explanation:

Step1: Find pairs with sum 7

The pairs of numbers from 1 - 5 that sum to 7 are (2,5), (3,4), (4,3), (5,2). So, the number of pairs with sum 7 is $n(A)=4$.

Step2: Find pairs with sum 7 and first - number 3 or higher

The pairs that sum to 7 and have the first number 3 or higher are (3,4), (4,3), (5,2). So, the number of such pairs is $n(A\cap B)=3$.

Step3: Use conditional - probability formula

The formula for conditional probability is $P(B|A)=\frac{P(A\cap B)}{P(A)}$. Since each outcome is equally - likely, and in the context of counting outcomes, $P(B|A)=\frac{n(A\cap B)}{n(A)}$. Substituting $n(A\cap B) = 3$ and $n(A)=4$ into the formula, we get $P=\frac{3}{4}$.

Answer:

D. $\frac{3}{4}$