A Certain Sports Car Comes Equipped with Either an Automatic

question 13

Essay

A certain sports car comes equipped with either an automatic or a manual transmission, and the car is available in one of four colors. Relevant probabilities for various combinations of transmission type and color are given in the accompanying table. $$\begin{array}{l}\begin{array} { c c c c c } &&&\text { Color }\\\hline \text { Transmission Type } & \text { White } & \text { Blue } & \text { Black } & \text { Red } \\\hline \text { A } & .13 & .10 & .11 & .11 \\\hline \mathrm { M } & .15 & .07 & .15 & .18 \\\hline\end{array}\end{array}$$ Let A = (automatic transmission), B = {black}, and C = {white}.
a. Calculate P(A), P(B), and P(A $\cap$
B).
b. Calculate both P(A|B) and P(B|A), and explain in context what each of these probabilities represents.

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Definitions:

3-Hexanone

A ketone with the formula C6H12O, characterized by a carbonyl group located on the third carbon atom of a six-carbon chain, used in fragrances and as a solvent.

Proton Spin Decoupled

A nuclear magnetic resonance (NMR) technique where the interaction between proton spins is removed or reduced, enhancing spectral resolution.

0-50 Ppm

A concentration range indicating 0 to 50 parts per million, often used to describe the presence of a substance within another substance.

Proton Spin Decoupled

A nuclear magnetic resonance (NMR) technique used to simplify the spectrum and analyze the structure by decoupling the interactions between proton spins.