Helmets and injuries the data in the accompanying table is...

Helmets and Injuries: The data in the accompanying table is based on data from 'Helmet Use and Risk of Head Injuries in Alpine Skiers and Snowboarders' by Sullheim et al., Journal of the American Medical Association, Vol. 295, No. Head Injuries Not Injured Wore Helmet 96 656 No Helmet 480 2330 What is the probability that a randomly selected person has a head injury? What is the probability that a randomly selected person wore a helmet? What is the probability that a randomly selected person had a head injury or wore a helmet? What is the probability that a randomly selected person did not wear a helmet and was not injured? What is the probability that a person did not wear a helmet, given they had a head injury? Are the events for helmet use and head injury independent? Prove mathematically:

Transcript

00:01 So i believe this, i have written down the data that you've given kind of in a linear fashion and a line.

00:07 I think this is how i'm interpreting it.

00:09 So if i've interpreted the way your data is given incorrectly, you will be able to see how you do the problem.

00:17 So i found the row totals and column totals.

00:20 And your first question is, if you randomly choose a person, what is the likelihood that the person is injured, has a head injury? and i called that this category.

00:31 And so there are two ways, either railing a helmet or not wearing a helmet.

00:35 So we have 576 out of 3562.

00:40 And that 3562 i have stored because we're going to use that on each, almost each question.

00:46 So 576 divided by that total gives us a probability of approximately 0 .1617.

00:53 I'll give them as a four -place decimal.

00:55 Now, the probability that someone was wearing a helmet, if you randomly choose a person, we have 752 people wearing a helmet in the data collection out of that 3562.

01:08 So 752 divided by that total, again, to a four -place decimal's point 2 -1 -1.

01:16 Now on number three, we want to find the probability of the injured or wearing a helmet.

01:21 And we know you can find that by taking the sum of the probability, so that five, 576 over that 3562 plus adding on the probability of wearing a helmet, which is 752, but we would have overlapped.

01:39 And which ones have we overlapped? well, we have helmet here and injury here.

01:45 We've double counted the intersections.

01:47 We have to subtract that away.

01:50 Let me get back to the writing.

01:52 So subtract away 96.

01:53 Make sure that that's 96.

01:54 Yes, it is.

01:55 So we have the that 576 plus the 752 and subtract away that overlap that we double counted.

02:03 And then that comes out to be 1, 2, 3, 2, 2 divided by that 3562.

02:09 And once again, i'll get that to you as a four -place decimal.

02:11 So that comes out to be approximately 0 .349.

02:16 That is for this question.