Virus Probabilities

 I saw something on the news about the coronavirus. Wearing a mask is 70% effective in blocking the droplets that can cause you to be sick. If you and the other person is wearing a mask, you only have a 9% of getting infected. (Those numbers are what I heard, but I'm not certain of the context because I wasn't totally paying attention, but we'll run with it.) Those numbers seem right, mathematically. You have person A and person B. You want either one's mask to block the germs.

Probablility of A or B is happening is 

Pr(A) + Pr(B) - Pr(A) x Pr(B).=  0.7  +  0.7  - 0.7 x 0.7 = 1.4 - 0.49 = 0.91. 

Thus,a 9% chance germs get through.

Looking at it another way, start directly with the probability germs get through.

C and D are now the events that germs aren't blocked.

Pr(C) x Pr(D) = 0.3 x 0.3 = 0.09

Again a 9% chance.

Pr(B) - Pr(A) x Pr(B).=  0.3 +  0.  - 0.7 x 0.7 = 1.4 - 0.49 = 0.91. 

Leaves

 I looked out upon my lawn covered with leaves and wondered how many there were. The brute force method is to go count them. Lets say I don't have the time or inclination to do that. I could take some smaller areas of the lawn, count the leaves there, and compare them to the whole.

This is the same kind of thing that is done in surveys. DVD sales, TV ratings, unemployment numbers, all survey to get their results. We recently had a presidential election. It took a while, a long while, because they said they were getting to a 99.5%. That doesn't give you certainty of a number. It just means you are 99.5% certain of being within a range of numbers. Almost all nationwide statistics are surveys. When we don't do that - the census every 10 years - it takes months and months to get to everyone.

I'm sure you know how this kind of thing works, but I took the time to do it, so I feel obligated to share.

I (roughly) marked one yard by one yard squares. There were three of them and they averaged having 125 leaves in each. Then I found the area of the entire yard. Nicely, it is pretty much a trapezoid. The bases about 7 and 14 yards with a height of 7 yards. That gives an area of 73.5 square yards.

Next, a proportion would be 125/1 = x/73.5. So, x = 9187.5 leaves. 

So, a nice problem for maybe an Algebra I or general math class. It can lead into the idea of the surveys that go on every day.