the numbers 1, 2, 3, 4, and 5 are written on slips of paper, and 2 slips are drawn at random one at a time…
the numbers 1, 2, 3, 4, and 5 are written on slips of paper, and 2 slips are drawn at random one at a time without replacement. find the given probabilities. a. the sum of the numbers is 8. b. the sum of the number is 4 or less. c. the first number is 2 or the sum is 5. a. the probability that the sum of the numbers is 8 is $\frac{1}{10}$ (type an integer or a simplified fraction.) b. the probability that the sum of the numbers is 4 or less is (type an integer or a simplified fraction.)
Answer
Explanation:
Step1: Calculate total number of outcomes
The number of ways to draw 2 - number combinations from 5 numbers without replacement is given by the permutation formula $P(n,r)=\frac{n!}{(n - r)!}$, where $n = 5$ and $r=2$. So, $P(5,2)=\frac{5!}{(5 - 2)!}=\frac{5!}{3!}=5\times4 = 20$ ways. But since the order of drawing does not matter for the sum - related problems, the number of combinations $C(5,2)=\frac{5!}{2!(5 - 2)!}=\frac{5\times4}{2\times1}=10$ combinations.
Step2: Find combinations for sum is 4 or less
The possible combinations of two numbers from ${1,2,3,4,5}$ whose sum is 4 or less are $(1,2)$ and $(1,3)$. So there are 2 combinations.
Step3: Calculate probability
The probability $P$ of an event is given by $P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. Here, the number of favorable outcomes is 2 and the total number of outcomes is 10. So $P=\frac{2}{10}=\frac{1}{5}$.
Answer:
$\frac{1}{5}$