Questions: Find P (musician is a female).

Transcript text: 3.4 Contingency Tables Use the following information to answer the next four exercises. Table 3.15 shows a random sample of musicians and hor they learned to play their instruments. \begin{tabular}{|l|l|l|l|l|} \hline Gender & Self-taught & Studied in School & Private Instruction & Total \\ \hline Female & 12 & 38 & 22 & 72 \\ \hline Male & 19 & 24 & 15 & 58 \\ \hline Total & 31 & 62 & 37 & 130 \\ \hline \end{tabular} Table 3.15 54. Find $P$ (musician is a female).

Solution

Solution Steps

To find the probability that a randomly selected musician is female, we need to divide the total number of female musicians by the total number of musicians. This will give us the probability of selecting a female musician from the sample.

Step 1: Identify the Total Number of Female Musicians

The total number of female musicians is given as 72.

Step 2: Identify the Total Number of Musicians

The total number of musicians, including both male and female, is given as 130.

Step 3: Calculate the Probability

The probability $ P $ that a randomly selected musician is female is calculated by dividing the total number of female musicians by the total number of musicians: \[ P(\text{female}) = \frac{\text{Number of Female Musicians}}{\text{Total Number of Musicians}} = \frac{72}{130} \]

Step 4: Simplify the Fraction

Simplifying the fraction $\frac{72}{130}$ gives: \[ P(\text{female}) = \frac{36}{65} \]

Step 5: Convert to Decimal

Converting the simplified fraction to a decimal: \[ P(\text{female}) \approx 0.5538 \]