Question

1. the whole numbers 1 through 30 were each written on separate pieces of paper. those 30 pieces of paper were put into a jar. one piece of paper will be randomly drawn from this jar. what is the probability that this piece of paper will have a prime number written on it?

f. $\frac{1}{30}$
g. $\frac{1}{20}$
h. $\frac{1}{10}$
j. $\frac{10}{30}$

1. for an angle with measure $\alpha$ in a right triangle, $\sin\alpha=\frac{5}{13}$ and $\tan\alpha=\frac{5}{12}$. what is the value of $\cos\alpha$?

a. $\frac{12}{13}$
b. $\frac{12}{\sqrt{194}}$
c. $\frac{12}{\sqrt{119}}$
d. $\frac{13}{12}$

1. which of the following values, if any, is the $y$-value of the solution set to the system of equations below?

$2x - y = 7$
$-4x + 2y = 2$
f. 2
g. 5
h. 9
j. there is no such value for $y$.

Explanation:

Step1: Recall the formula for probability

The probability \(P\) of an event is given by \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). Here, the total number of outcomes is \(n = 30\) (since there are 30 pieces of paper numbered from 1 to 30).

Step2: List prime numbers between 1 and 30

Prime numbers are numbers greater than 1 that have only two distinct positive divisors: 1 and the number itself. The prime numbers between 1 and 30 are \(2,3,5,7,11,13,17,19,23,29\). The number of prime numbers (favorable outcomes) \(m=10\).

Step3: Calculate the probability

Using the probability formula \(P=\frac{m}{n}\), substitute \(m = 10\) and \(n=30\). So, \(P=\frac{10}{30}\).

Answer:

J. \(\frac{10}{30}\)