Two-Way Frequency Table & Joint Probability Solution Guide

Store Inventory Overview: Take Your Pick

• Store Name: Take Your Pick
• Total Instrument Inventory ($N$): $64$ total instruments (guitars and pianos combined)
• Instrument Categories:
  • Instrument Types: Guitars ($G$) and Pianos ($P$)
  • Power/Sound Types: Electric ($E$) and Acoustic ($A$)

Given Inventory Data

• Total Electric Instruments ($N(E)$): $36$
• Total Pianos ($N(P)$): $29$
• Acoustic Pianos ($N(P \text{ and } A)$): $12$

Two-Way Frequency Table Construction

To find all missing quantities across the inventory categories, step-by-step arithmetic deductions are applied:

• Total Acoustic Instruments ($N(A)$):$N(A) = \text{Total Instruments} - N(E) = 64 - 36 = 28$

• Total Guitars ($N(G)$):$N(G) = \text{Total Instruments} - N(P) = 64 - 29 = 35$

• Electric Pianos ($N(P \text{ and } E)$):$N(P \text{ and } E) = N(P) - N(P \text{ and } A) = 29 - 12 = 17$

• Electric Guitars ($N(G \text{ and } E)$):$N(G \text{ and } E) = N(E) - N(P \text{ and } E) = 36 - 17 = 19$

• Acoustic Guitars ($N(G \text{ and } A)$):$N(G \text{ and } A) = N(G) - N(G \text{ and } E) = 35 - 19 = 16$$\text{Check: } N(A) - N(P \text{ and } A) = 28 - 12 = 16$

Complete Contingency Table

Probability Analysis: Piano and Electric Instrument

Target Event

• Determining the probability that a randomly selected instrument from the store is both a piano and electric ($P(\text{Piano and Electric})$).

Mathematical Formula

$P(\text{Piano and Electric}) = \frac{N(\text{Piano and Electric})}{\text{Total Inventory}}$

Calculation

• Favorable Outcomes ($N(P \text{ and } E)$): $17$
• Sample Space ($N$): $64$

$P(\text{Piano and Electric}) = \frac{17}{64}$

Equivalent Representations

• Simplified Fraction: $\frac{17}{64}$
• Decimal Value: $0.265625$
• Percentage: $26.5625\text{\%}$ or approximately $26.56\text{\%}$

Related Marginal and Conditional Probabilities

• Probability of Selecting Any Piano:$P(\text{Piano}) = \frac{29}{64} \thickapprox 0.4531 \quad (45.31\text{\%})$

• Probability of Selecting Any Electric Instrument:$P(\text{Electric}) = \frac{36}{64} = \frac{9}{16} = 0.5625 \quad (56.25\text{\%})$

• Conditional Probability that a Piano is Electric ($P(\text{Electric} \thickspace | \thickspace \text{Piano})$):$P(\text{Electric} \thickspace | \thickspace \text{Piano}) = \frac{N(P \text{ and } E)}{N(P)} = \frac{17}{29} \thickapprox 0.5862 \quad (58.62\text{\%})$

• Conditional Probability that an Electric Instrument is a Piano ($P(\text{Piano} \thickspace | \thickspace \text{Electric})$):$P(\text{Piano} \thickspace | \thickspace \text{Electric}) = \frac{N(P \text{ and } E)}{N(E)} = \frac{17}{36} \thickapprox 0.4722 \quad (47.22\text{\%})$