question about additive benefit of everyone wearing masks during covid19

[eric beans]

I was listening to a podcast where someone was saying that even if wearing bandanna/homemade masks were only 20% effective, if everyone wore it, it would at least be 80% effective. They mentioned the additive nature.

I was puzzled how they got to 80%. I'm thinking if 2 people are wearing masks and say the bandanna are only 20% effective, wouldn't that only add up to 40%? What am I missing here?

 

[Dr.Peterson]

I'd have to see the actual argument.

As I see it, two would not even be 40% effective. If each masks blocks 20%, then it lets through 80% of the virus. If the same is true for both breathing in and out, then two people with masks would transmit 80% of 80%, which is 64%; so I'd call them 36% effective. Nothing here is really additive.

Maybe they have something more than just two people in mind. I can't think of any way to come to their conclusion, though.

 

[Harry_the_cat]

Lies, damned lies and statistics!

 

[eric beans]

I heard it on Bret Weinstein's 5th podcast.

at 52:26. Listen for yourself. Maybe I didn't quite understand what he was saying....

 

[Romsek]

using Dr. Peterson's approach we can make a graph of "effectiveness" as a function of the number worn.

[MATH]e = 1 - (1-p)^n\\ \text{where $e$ is the overall effectiveness and $p$ is the effectiveness of a single mask}[/MATH]


and we see it actually takes about 7 masks to reach 80% effectiveness.

 

[Dr.Peterson]

I heard it on Bret Weinstein's 5th podcast. ... at 52:26. Listen for yourself. Maybe I didn't quite understand what he was saying....

I don't think he's done any math at all, just made an unsupported guess.

 

[eric beans]

using Dr. Peterson's approach we can make a graph of "effectiveness" as a function of the number worn.

[MATH]e = 1 - (1-p)^n\\ \text{where $e$ is the overall effectiveness and $p$ is the effectiveness of a single mask}[/MATH]
View attachment 17776

and we see it actually takes about 7 masks to reach 80% effectiveness.

Um, could you explain how they derived e = 1 - (1-p)^n???? I don't understand.

 

[eric beans]

I don't think he's done any math at all, just made an unsupported guess.

So his notion that bandannas are effective even if they are only 20% effective is baseless? Correct? I guess the only truly effective mask is N95 and those wearing bandannas look like silly highway robbers from westerns.

 

[Steven G]

I was listening to a podcast where someone was saying that even if wearing bandanna/homemade masks were only 20% effective, if everyone wore it, it would at least be 80% effective. They mentioned the additive nature.

I was puzzled how they got to 80%. I'm thinking if 2 people are wearing masks and say the bandanna are only 20% effective, wouldn't that only add up to 40%? What am I missing here?

Let's assume that everything you said is true. Your logic with your conclusion is totally flawed. It was said that if ALL people wore masks then it would be 80% effective (again, lets assume this is all true). Well if only two people out of all people wore masks don't you think that the effectiveness would be less than if everyone wore masks? Since 40% is less than 80% I do not understand why you are surprised with the result of 40% effectiveness.

For the record that report is flawed. Dr Peterson, a mathematician, is correct when he says that the two masked people would receive 36% effectiveness. The reason I stated that Dr Peterson is a mathematician is because personally I would not believe what anyones say about the spread of the Corona Virus except for a mathematician or a statistician and I might have trouble believing the statistician.

 

[Steven G]

So his notion that bandannas are effective even if they are only 20% effective is baseless? Correct? I guess the only truly effective mask is N95 and those wearing bandannas look like silly highway robbers from westerns.

No, I don't think that anyone is saying this. I would prefer to see everyone wearing bandanas compared to 10% of people wearing N95 masks.

On another note. If you do come into a N95 mask you should not use it for yourself, instead you should donate it to a hospital.

 

[eric beans]

I'd have to see the actual argument.

As I see it, two would not even be 40% effective. If each masks blocks 20%, then it lets through 80% of the virus. If the same is true for both breathing in and out, then two people with masks would transmit 80% of 80%, which is 64%; so I'd call them 36% effective. Nothing here is really additive.

Maybe they have something more than just two people in mind. I can't think of any way to come to their conclusion, though.

Excuse my ignorance guys, this equation, e = 1 - (1-p)^n, i don't understand where this equation came from. Could someone derive this? Could you explain how to get 64% from the 80% and 80%? This seems simple but I'm still not getting this. Thanks guys.

 

[Steven G]

% means to divide by 100
of means to multiply

In my work below I will keep one of the % signs because I want my answer to have a % sign

[math]80\% \ of \ 80\% = \dfrac{80}{100}*80\% = \dfrac{6400}{100}\% = 64\%[/math].


If a $80 item is on sale at 20% off (that means you pay 80%) how do you find the sale price?

 

[Steven G]

Excuse my ignorance guys, this equation, e = 1 - (1-p)^n, i don't understand where this equation came from. Could someone derive this?

p = chance of a mask being effective = .20. Then 1-p = 1-.20 = .80 = the chance that the mask is ineffective. (1-p)^n is the chance that if n people wore masks it would be ineffective. Then 1-(1-p)^n = 1-(.8)^n is the chance that if n people wore masks it would be effective

 

[eric beans]

I don't understand why this would multiplicative (exponential) rather than additive effect?

 

[Dr.Peterson]

Because percentage effectiveness is inherently multiplicative! "20% of ..." means "0.20 times ..."

Moreover, "20% effective" means "removes 20% of x", so the result is, as Jomo said, "x - 0.20x", which is "0.80x". So each time air is filtered through this, the amount of virus is multiplied by 0.80.

 

[eric beans]

p = chance of a mask being effective = .20. Then 1-p = 1-.20 = .80 = the chance that the mask is ineffective. (1-p)^n is the chance that if n people wore masks it would be ineffective. Then 1-(1-p)^n = 1-(.8)^n is the chance that if n people wore masks it would be effective

why don't they just use (.2)^n to figure out the effectiveness? why did they have to use ineffectiveness (1-p)^n and then subtract that from total of 1 to get at the effectiveness? it seems rather round about way to get at the effectiveness of the masks?

 

[Dr.Peterson]

why don't they just use (.2)^n to figure out the effectiveness? why did they have to use ineffectiveness (1-p)^n and then subtract that from total of 1 to get at the effectiveness? it seems rather round about way to get at the effectiveness of the masks?

If a mask filters out 20% of the virus, then it lets through 80%. That 80% is what the next person's mask has to deal with! So the second mask blocks out 20% of the 80%, letting through 80% of the 80%, or 64%. Thus, 36% are blocked out. This is what I said in post #2. (But note that this is just a small decrease from what you get by adding.)

Now, I disagree with the idea introduced in #5, using a higher power than 2, because between any two people there are only the two masks, not the masks worn by everyone else. The reason this has to do with everyone wearing a mask is that if any don't, then those people have a greater exposure. So although the formula with n is a valid formula in probability, I don't think this is a probability question.

But there are other issues here. One is whether wearing masks reduces the viral load, reducing the chance of getting sick, or its severity, by more than 35%. Everything is not necessarily linear.