Question

a survey was taken of automobile preferences for different age group. they were asked whether they would prefer owning a car or a truck. find each probability as a fraction.

	truck	car	total
36 - 55	30	160	190
total	50	200	250

1. p(car) =
2. p(truck | 36 - 55) =
3. p(16 - 35 and car) =

Explanation:

Step1: Find total number of respondents

The total number of respondents is 250 (from the grand - total in the table).

Step2: Calculate \(P(\text{Car})\)

The number of people who prefer cars is 200. So \(P(\text{Car})=\frac{\text{Number of people who prefer cars}}{\text{Total number of respondents}}=\frac{200}{250}=\frac{4}{5}\).

Step3: Calculate \(P(\text{Truck}|\text{36 - 55})\)

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). In terms of the table, \(P(\text{Truck}|\text{36 - 55})=\frac{\text{Number of 36 - 55 year - olds who prefer trucks}}{\text{Total number of 36 - 55 year - olds}}\). The number of 36 - 55 year - olds who prefer trucks is 30, and the total number of 36 - 55 year - olds is 190. So \(P(\text{Truck}|\text{36 - 55})=\frac{30}{190}=\frac{3}{19}\).

Step4: Calculate \(P(\text{16 - 35 and car})\)

The number of 16 - 35 year - olds who prefer cars is 40, and the total number of respondents is 250. So \(P(\text{16 - 35 and car})=\frac{\text{Number of 16 - 35 year - olds who prefer cars}}{\text{Total number of respondents}}=\frac{40}{250}=\frac{4}{25}\).

Answer:

1. \(\frac{4}{5}\)
2. \(\frac{3}{19}\)
3. \(\frac{4}{25}\)