Question

finding probability: guided practice
each face of a fair 14 - sided die is labeled with a number from 1 through 14, with a different number appearing on each face. if the die is rolled one time, what is the probability of rolling a 2?
a. $\frac{1}{14}$
b. $\frac{3}{14}$
c. $\frac{12}{14}$
d. $\frac{13}{14}$
a bag contains 8 red marbles, 9 yellow marbles, and 7 green marbles. how many additional red marbles must be added to the 24 marbles already in the bag so that the probability of randomly drawing a red marble is $\frac{3}{5}$?
a. 11
b. 16
c. 20
d. 24
e. 32

Explanation:

First Question (14 - sided die probability)

Step1: Identify total outcomes

A 14 - sided die has 14 possible outcomes (numbers 1 - 14), so total outcomes \( n = 14 \).

Step2: Identify favorable outcomes

We want to roll a 2, so there is 1 favorable outcome (\( m = 1 \)).

Step3: Calculate probability

Probability \( P=\frac{\text{favorable outcomes}}{\text{total outcomes}}=\frac{m}{n}=\frac{1}{14} \).

Step1: Define variables

Let \( x \) be the number of additional red marbles. Initial red marbles \( r = 8 \), total marbles initially \( T = 24 \). After adding \( x \) red marbles, red marbles \( r'=8 + x \), total marbles \( T'=24 + x \).

Step2: Set up probability equation

The probability of drawing a red marble is \(\frac{r'}{T'}=\frac{3}{5}\), so \(\frac{8 + x}{24 + x}=\frac{3}{5}\).

Step3: Cross - multiply and solve

Cross - multiply: \( 5(8 + x)=3(24 + x) \)
Expand: \( 40+5x = 72 + 3x \)
Subtract \( 3x \) from both sides: \( 40 + 2x=72 \)
Subtract 40 from both sides: \( 2x=32 \)
Divide by 2: \( x = 16 \)

Answer:

A. \(\frac{1}{14}\)

Second Question (Marble probability)