two integers are randomly selected from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} and are added together. find the…
two integers are randomly selected from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} and are added together. find the probability that their sum is 5 if they are selected (a) with replacement, (b) without replacement. (a) the probability that their sum is 5 if they are selected with replacement is . (type an integer or a simplified fraction.)
Answer
Explanation:
Step1: Calculate total outcomes with replacement
The set has 9 elements. When selecting two integers with replacement, the total number of outcomes is (n = 9\times9=81) since for the first - number there are 9 choices and for the second number there are also 9 choices.
Step2: Find pairs that sum to 5 with replacement
The pairs ((x,y)) such that (x + y=5) and (x,y\in{1,2,3,4,5,6,7,8,9}) are ((1,4)), ((2,3)), ((3,2)), ((4,1)). So there are (m = 4) favorable outcomes.
Step3: Calculate probability with replacement
The probability (P=\frac{m}{n}=\frac{4}{81}).
Step4: Calculate total outcomes without replacement
When selecting two integers without replacement, the total number of outcomes is (n=\text{C}_{9}^2=\frac{9!}{2!(9 - 2)!}=\frac{9\times8}{2\times1}=36).
Step5: Find pairs that sum to 5 without replacement
The pairs ((x,y)) such that (x + y = 5) and (x,y\in{1,2,3,4,5,6,7,8,9}) without replacement are ((1,4)), ((2,3)). So there are (m = 2) favorable outcomes.
Step6: Calculate probability without replacement
The probability (P=\frac{m}{n}=\frac{2}{36}=\frac{1}{18}).
Answer:
(a) (\frac{4}{81}) (b) (\frac{1}{18})