Are They Wearing Seatbelts?
Date: 26 Mar 1995 06:27:58 -0500 From: Anonymous Subject: Seatbelts 80% of all California drivers wear seatbelts. If 4 drivers were pulled over, what is the probability that all 4 would be wearing their seat belts? What is it that none of them would be wearing their seatbelts? Thanx
Date: 26 Mar 1995 14:10:04 -0500 From: Elizabeth Weber Subject: Re: Seatbelts Hello there! The number of drivers in California is large enough that pulling one over won't affect the % left who are wearing seatbelts. This makes your problem pretty straightforward. The probability that any driver we pull over is wearing a seatbelt is 80% The probability that any second driver we pull over is also wearing a seatbelt is also 80%, so the probability that both of the drivers we pull over are wearing seat belts is 80% of 80%, or .8 x .8. To see why this is so, you can try drawing something called a Venn diagram. Take a piece of paper, and mark it so that it's divided into two pieces, of which one takes up 80% of the paper, and the other takes up 20%, something like this: __________________ | 20% | --probability that the driver |__________________| is not wearing a seat belt | | | | | | | 80% | --probability that the driver | | is wearing a seat belt | | | | |__________________| This is how the probability of pulling over a driver wearing a seatbelt looks after you've pulled over one driver. Now, let's say you've pulled over a driver wearing a seatbelt, and you want to see what the probability of pulling over a second driver who is wearing a seatbelt looks like. You're already inside the piece of your paper which takes up 80% of the paper, so you divide this piece into 2 smaller pieces, just like you divided the entire piece of paper before: __________________ | 20% | |__________________| | | | | | | | 80%| | | x | 80% x 80%| | 20%| | | | | | | | |_______|__________| The 80% x 80% box is the box you're interested in. Now, can you see how you would divide it to find the probability that the third driver you pull over is wearing a seatbelt if the first two you've pulled over were wearing seatbelts? Can you see how to find the probability of pulling over 4 drivers in a row who are all wearing seatbelts? Can you see how to find the probability that none of the drivers you pull over are wearing seatbelts? If you need more help, just let us know. Thanks for writing to Dr. Math! Elizabeth, a math doctor
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