Question

frequency tables, probability, and independent events
the two - way frequency table shows the current inventory of hardwood that a lumberyard carries. the hardwood board dimensions are given (e.g., 1×2 means \1 foot by 2 feet\) along with their types.
study the worked example and then answer each question.
suppose a board is chosen at random from the inventory, and it is a 1×2 board. does this choice affect the probability of choosing a maple board from the inventory?
p(maple) =
p(1×2) =
p(maple and 1×2) =

Explanation:

Step1: Calculate \( P(\text{maple}) \)

The formula for probability is \( P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \).
The total number of boards is \( 210 \), and the number of maple boards is \( 70 \).
So, \( P(\text{maple})=\frac{70}{210}=\frac{1}{3} \).

Step2: Calculate \( P(1\times2) \)

The number of \( 1\times2 \) boards is \( 42 \), and the total number of boards is \( 210 \).
Using the probability formula \( P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \), we get \( P(1\times2)=\frac{42}{210}=\frac{1}{5} \).

Step3: Calculate \( P(\text{maple and }1\times2) \)

The number of maple \( 1\times2 \) boards is \( 14 \), and the total number of boards is \( 210 \).
By the probability formula \( P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \), \( P(\text{maple and }1\times2)=\frac{14}{210}=\frac{1}{15} \).

Answer:

\( P(\text{maple})=\frac{1}{3} \)
\( P(1\times2)=\frac{1}{5} \)
\( P(\text{maple and }1\times2)=\frac{1}{15} \)