Conditional probability

1. Dec 16, 2013

DotKite

1. The problem statement, all variables and given/known data

A study of texting and driving has found that 40% of all fatal auto accidents
are attributed to texting drivers, 1% of all auto accidents are fatal, and
drivers who text while driving are responsible for 20% of all accidents. Find
the percentage of non-fatal accidents caused by drivers who do not text.

2. Relevant equations

3. The attempt at a solution

Let T denote texting while driving and let F denote fatal accidents.

P(F|T) = .40
P(F) = .01
P(T) = .2

I guess we are trying to find
p(F$^{c}$|T$^{c}$
= (p(F$^{c}$$\bigcap$ T$^{c}$) / p(T$^{c}$

We know p(F|T) = p(F$\bigcap$T) / p(T) = 0.4 => p(F$\bigcap$T) = 0.08

Also p(F$^{c}$ $\bigcap$T$^{c}$) = 1 - p(F$\bigcup$T)

p(F$\bigcup$T) = p(F) + p(T) - p(F $\bigcap$T) = .01 + .2 - .08 = .13

I am gonna stop here because when I start plugging in everything I have I wind up with the wrong answer. Is ther an assumption I have wrong or have interpreted, as usual, the problem wrong?

2. Dec 16, 2013

Staff: Mentor

That would be "For texting drivers, 40% of all accidents are fatal", which does not match the problem statement.

3. Dec 16, 2013

DotKite

So would it be p(F$\bigcap$T) = .40?

You got any hints?

Last edited: Dec 16, 2013
4. Dec 16, 2013

Dick

Now that reads "in 40% of all accidents the accident was fatal and the driver was texting". Still not what you want, given the probability of a fatal accident is only 0.01. Try again. Read these probability statement back in english. Here's a big hint. What does P(T|F) mean? State it in english.

5. Dec 16, 2013

DotKite

P(T|F) reads the probability of an accident being caused by texting, given that it was fatal?

6. Dec 16, 2013

Dick

Ok, but, you should read it a little more literally. Nobody said anything about texting being the cause. It is just the probability that driver was texting given the accident was fatal. Now what's the value of that given the problem statement?

7. Dec 17, 2013

DotKite

0.4?

8. Dec 17, 2013

HallsofIvy

Imagine 1000 accidents. "1% of all auto accidents are fatal" so there are 10 fatal accidents. "40% of all fatal auto accidents are attributed to texting drivers" so 4 of those fatal accidents are attributable to texting. "drivers who text while driving are responsible for 20% of all accidents" so 200 accidents are attributable to drivers who text.

That is, out of 200 accidents attributable to drivers who text, 4 of them are fatal and 16 are not fatal.

9. Dec 17, 2013

Dick

Yes, there's a big difference between P(T|F) and P(F|T). You might want to take another look at the expression you wrote for what you are trying to find.

10. Dec 17, 2013

haruspex

Typo (?):
... and 196 are not fatal.

11. Dec 18, 2013

HallsofIvy

Yes, thanks. Unfortunately, I can no longer edit it so I cannot pretend I didn't make that blunder!