12.4: simple probability and sample spaces discussion blake saved up to buy a vehicle in cash. he has a…

12.4: simple probability and sample spaces discussion blake saved up to buy a vehicle in cash. he has a choice of getting a sedan, truck, or van with leather or fabric interior, and a gps and/or sunroof. his friend tells him he has a total of 12 options. is his friend correct? why or why not? explain your reasoning.

12.4: simple probability and sample spaces discussion blake saved up to buy a vehicle in cash. he has a choice of getting a sedan, truck, or van with leather or fabric interior, and a gps and/or sunroof. his friend tells him he has a total of 12 options. is his friend correct? why or why not? explain your reasoning.

Answer

Explanation:

Step1: Count vehicle - type options

There are 3 vehicle - type options (sedan, truck, van), so $n_1 = 3$.

Step2: Count interior - type options

There are 2 interior - type options (leather, fabric), so $n_2=2$.

Step3: Count feature - combination options

For the GPS and sun - roof features, the possible combinations are: no GPS and no sun - roof, GPS only, sun - roof only, GPS and sun - roof. So there are 4 combinations, $n_3 = 4$.

Step4: Use the counting principle

The total number of options is found by multiplying the number of options for each category together. Using the multiplication principle of counting, the total number of options $N=n_1\times n_2\times n_3$. Substitute $n_1 = 3$, $n_2 = 2$, and $n_3 = 4$ into the formula: $N=3\times2\times4=24$.

Answer:

No, his friend is not correct. The total number of options is 24, not 12, because there are 3 vehicle - type options, 2 interior - type options, and 4 combinations of features (GPS and sun - roof), and using the multiplication principle of counting $3\times2\times4 = 24$.