Example 3.20 from Statistics, 3.4 Contingency Tables

Contingency Tables: example 3.20

Find P(Person uses a cell phone while driving).
Find P(Person had no violation in the last year).
Find P(Person had no violation in the last year and uses a cell phone while driving).
Find P(Person uses a cell phone while driving or person had no violation in the last year).
Find P(Person uses a cell phone while driving given person had a violation in the last year).
Find P(Person had no violation last year given person does not use a cell phone while driving).

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
a. This is the same as the marginal distribution (Section 1.2).
$P (Person uses a cell phone while driving) = \frac{number who use cell phones while driving}{number in study} = \frac{305}{755} \approx .4040$
3.5

b. The marginal distribution is

$P (Person had no violation in the last year) = \frac{number who had no violation}{number in study} = \frac{685}{755} \approx .9073 .$
3.6

c. Find the number of participants who satisfy both conditions.

$\begin{matrix} P (P e r s o n h a d n o v i o l a t i o n i n t h e l a s t y e a r A N D u s e s a c e l l p h o n e w h i l e d r i v i n g) = \frac{n u m b e r w h o h a d n o v i o l a t i o n A N D u s e s c e l l p h o n e w h i l e d r i v i n g}{n u m b e r i n s t u d y} \\ = \frac{280}{755} \\ \approx .3709 \end{matrix}$
3.7

d. To find this probability, you need to identify how many participants use a cell phone while driving OR have no violation in the past year OR both.

$P (Person uses a cell phone while driving OR had no violation in the last year) = \frac{25 + 405 + 280}{755}$
$\begin{matrix} = \frac{710}{755} \\ \approx .9404 \end{matrix}$
3.8

e. This is a conditional probability. You are given that the person had no violation in the last year, so you need only consider the values in that column of data.

$\begin{matrix} (P e r s o n u s e s a c e l l p h o n e w h i l e d r i v i n g G I V E N t h e p e r s o n h a d a v i o l a t i o n i n t h e l a s t y e a r) = \frac{n u m b e r w h o u s e d c e l l p h o n e A N D h a d a v i o l a t i o n}{n u m b e r i n s t u d y w h o h a d a v i o l a t i o n i n t h e l a s t y e a r} \\ = \frac{25}{70} \\ \approx .3571 \end{matrix}$
3.9

f. For this conditional probability, consider only values in the row labeled “Does not use a cell phone while driving.”

$P (Person had no violation last year GIVEN person does not use cell phone while driving) = \frac{405}{450} = .9$

How did it go?

Solution

Solution