# Probability (1 Viewer)

#### frmldhyd

##### New Member
omg i feel stupid for asking this question cause I cant get it, im probs tired or something, but here it is anyways:

One layer of tinting material on a window cuts out 1/5 of the sun's UV rays.
1)what fraction would be cut out by using two layers?
2) How many layers would be required to cut out at least 9/10 of the sun's UV rays?

Answers:
1)9/25
2)11

SHOWS me workings!! fanks

#### funnytomato

##### Active Member
1) after 1 layer , we have 1-0.2 =0.8 passing thru
so 0.8 get to travel to the 2nd layer, where another 0.2 will be cut out

2) so it works in the same way as compound interest , only difference is that it "depreciates"
and a similar example would be
if the interest rate is 5% p a, how many years would a principal of P double if the interest if compounded annually ?

so P*(1.05) is the money after 1 year, P*(1.05)*(1.05) is the money after another year
after n years , we have P*1.05^n which should equal 2*P
so 1.05^n=2
take log of both sides
ln (1.05^n)= ln2
n *ln 1.05= ln2
n = ln2 / ln 1.05 = 14.2066...=15 for it to double as it only calculates interest annually

The question you're asking is very similar, but it has inequality in it
so be carefully with the inequality signs

#### funnytomato

##### Active Member
also if u haven't learned log , sub number in and see when it satisfies the inequality

which sucks when the number is large

#### Trebla

##### Administrator
Administrator
omg i feel stupid for asking this question cause I cant get it, im probs tired or something, but here it is anyways:

One layer of tinting material on a window cuts out 1/5 of the sun's UV rays.
1)what fraction would be cut out by using two layers?
2) How many layers would be required to cut out at least 9/10 of the sun's UV rays?

Answers:
1)9/25
2)11

SHOWS me workings!! fanks
This is a series type question, not probability.
Let x be the amount of the sun's rays that hits the material
First layer takes out x/5 so after the first layer there is only 4x/5 left that is not cut
When it hits the second layer, only 4x/5 of rays pass through and a fifth of that is removed
i.e. (4x/5)(1/5) is removed
Total amount cut = x/5 + 4x/25 = 9x/25 so fraction of rays cut is 9/25

Note that after each layer we multiply the amount of rays that entered by 4/5 since 1/5 is removed.
In terms of proportions:
After 1st layer the fraction of rays left is (4/5)
After 2nd layer the fraction of rays left is (4/5)2
After 3rd layer the fraction of rays left is (4/5)3
...
After nth layer the fraction of rays left is (4/5)n

We want the fraction of rays left after the nth layer to be at least 1/10 (as 9/10 is cut out) so
(4/5)n > 9/10
Using logarithms or otherwise this should yield n = 11 as the lowest integer which satisfies this inequality

Thanks!!

#### funnytomato

##### Active Member
This is a series type question, not probability.
LOL, I felt that something was not right

#### shmitler

##### New Member
This is a series type question, not probability.
Let x be the amount of the sun's rays that hits the material
First layer takes out x/5 so after the first layer there is only 4x/5 left that is not cut
When it hits the second layer, only 4x/5 of rays pass through and a fifth of that is removed
i.e. (4x/5)(1/5) is removed
Total amount cut = x/5 + 4x/25 = 9x/25 so fraction of rays cut is 9/25

Note that after each layer we multiply the amount of rays that entered by 4/5 since 1/5 is removed.
In terms of proportions:
After 1st layer the fraction of rays left is (4/5)
After 2nd layer the fraction of rays left is (4/5)2
After 3rd layer the fraction of rays left is (4/5)3
...
After nth layer the fraction of rays left is (4/5)n

We want the fraction of rays left after the nth layer to be at least 1/10 (as 9/10 is cut out) so
(4/5)n > 9/10
Using logarithms or otherwise this should yield n = 11 as the lowest integer which satisfies this inequality
i believe you have a made a mistake but please correct me if i am wrong
where you wrote (4/5)^n>9/10, it is meant to be (4/5)^n<1/10 as the number of layers you want to cut out is 9/10 and leave 1/10 light remaining
thanks • Roy G Biv