Question

akita wants to purchase that all - important first new car and can select one option from each category:
draw a tree diagram to determine the sample space and find the probabilities.
find the sample space for all possible choices. for the car model, ff means ford focus and hc means honda civic; for engine type, h means hybrid and e means e - 85; for color, bc means burnt copper and cb means cobalt blue.
{ffhbc, ffhcb, ffebc, ffecb, hchbc, hchcb, hcebc, hcecb}
{ffbc, ffcb, ff, ffe, ffh, hcbc, hccb, hc, hce, hch, tcbc, tccb, tc, tce, tch}
{ffhbc, ffhcb, ffh, hchbc, hchcb, ffh, tchbc, tchcb, tch}
{ffhbc, ffhcb, ffh, ffhbc, ffhcb, ffh, hchbc, hchcb, ffh, tchbc, tchcb, tch}
{ffhbc, ffhcb, ffh, ffhbc, ffhcb, ffh, hchbc, hchcb, hch, hcbc, hcbce, tchbc, tchcb, tch, tcbcb, tcebc, tcecb, tce, tcbcbce}
{ffhbc, ffhcb, ffh, ffhbc, ffhcb, ffh, hchbc, hchcb, hch, hcbc, hcbce, hchbc, ffhbc, ffh, ffhbc, ffhcb, ffh, hchbc, hchcb, ffh, tchbc, tchcb, tch}

Explanation:

Step1: Determine the number of choices for each category

• Model: There are 2 choices (Ford Focus and Honda Civic).
• Engine type: There are 2 choices (Hybrid and E - 85).
• Color: There are 2 choices (Burnt copper and Cobalt blue).

Step2: Use the multiplication principle

The multiplication principle states that if there are \(m\) ways to do one thing, \(n\) ways to do a second thing, and \(p\) ways to do a third thing, then there are \(m\times n\times p\) ways to do all three things together.
Here, \(m = 2\) (models), \(n=2\) (engine types), \(p = 2\) (colors).
The total number of elements in the sample space \(N=2\times2\times2=8\)

Step3: List all possible combinations

Let \(F\) represent Ford Focus, \(H\) represent Honda Civic, \(B\) represent Burnt copper, \(C\) represent Cobalt blue, \(Y\) represent Hybrid, \(E\) represent E - 85.
The sample space \(\Omega=\{FYB,FYC,FEB,FEC,HYB,HYC,HEB,HEC\}\)

Answer:

The sample space is \(\{FYB,FYC,FEB,FEC,HYB,HYC,HEB,HEC\}\)