IIT JEE , AIEEE Forum - Mathematics , Algebra - find probablity that colour of the traffic light changes in any random 3 sec interval

muditparnami

traffic light runs through following cycle green for 30 sec then yellow for 3sec and red for 30 sec. a random 3sec interval is chosen to watch the light

joyfrancis

6/91?

sboosy

Consider situation to start from the 1st second of red
red stands for 30 seconds ....1 2 3 4 ...30 - Red
then green ....31 32 ......60 - Green
then yellow ....61 62 63 - yellow
after this the same cycle is repeated
fav cases
(29 30 31 )
(30 31 32)
(59 60 61)
(60 61 62)
(62 63 64(next round 1))
(63 64(second round 1) 65 (Second round 2))
fav cases = 6
total 3 second interval in this
(1 2 3)
(2 3 4)
...
...
(63 64 65)
so 63
so ans = 6/63
= 2 /21
pls correct me if wrong

elastiboysai

The ans gvn by sboosy is rt, but his sample space is incomplete!!

so just for a more rigorous treatment of the qn

Assume we start observation at t=0

Choose your time interval along y axis, any arbitary time interval of length l.

we can observe signal change at green 2 yellow and at yellow to red

Any way length of time interval is fixed and 3 seconds, say we choose the time x.

We can see the change at green to yellow if x>27 and x<30

so this represents area l*3 a rectangle

now we can see the other color change if x>30 and and x<33

this again represents area l*3 a rectangle

so total favorable sample space is 6*l where u choose length of time of observation.

And total sample space is 63*l

so probability is 6l/63l=

2/21

This is assuming u choose length of time interval includes all 3 possibilities and is an integral multiple of 3.

muditparnami

sorry people but correct answer is 1/7 how let me explain hope u understand
since it is cycle consider a round table with 63 chairs first 30 belong to red section then next 30 to green then remaining three to yellow
now total no of three sec interval that we can choose are 63 (123,234,345.....63 1 2)
therefore sample space is 63 now fav cases are nine
(61 62 63)
62 63 1
63 1 2
28 29 30 (since at the end of 30 second red will change to green thus this is also fav case)
29 30 31
simlarly u can count for green yellow junction
therefor 9/63=1/7

sboosy

I am so very sorry to say ..that u can nothing like
"at the end of 30 it changes to" and all such things
that can no way b considered as a favourable case
so u reject three cases in each of ur colours
thus the answer is 6 / 63
now to b more clear
acoording to ur example . . assume the persons are wearing that colour shirts
u mean to say that going from 61 62 63 u change colour ???
all are wearing the same colour!!
and please dont say things as "since at the end it changes"
that is wrong

muditparnami

now listen sboosy if u were there standing in front of light on dot 30th sec after red light will turn to green right now (30-h)limit h tends to zero it is red on 30 th it is red thus it is a fav case sorry but i disagree

sboosy

now consider this ..
suppose u r to watch the a seconds clock closely and count how many seconds have passed by
u start when it is on 1 and i tell u to stop when it is 60
now wud u use the same limit expression of urs and say
(h tends to 0 ) and thus as i stopped seeing it became 61 and as i was just handing u the watch it became 62 and thus wud ur answer be 61 or 62 ..?
it shud be neither
the answer is a simple 60
u dont say tends to and all such things ..

elastiboysai

Hi Mudit parnami

I think both u n sboosy are forgettin that there are milliseconds between seconds and microseconds and nanoseconds

For ur info

this is a problem wid infinite sample space!!

U have messed up in listin d sample space

Ur analogy of chairs and cycle is totally wrong

as u see I can choose a time interval 27.0001 - 30.0001

27.000000000001 -30.000000000001

and obviously

ur sample space doesn take care of these!!

so weres da mistake

as u can see my method using areas includes all such time intervals( as it includes all points)

sboosy

Well elastic ...the question was obviously referring to human perceivable or more accurately countable seconds ...
suppose now a question appears in JEE
How many 3 second intervals are there between
12 and 12:10 (some time for example)
given the options ..wud u say sample space is infinite ..and leave the question ...
for if u go by ur argument ..there is no end is there?

elastiboysai

Well obviously sboosy

JEE wuldn have such a stupid question as the answer is, infinitely many.

Cmon yaar, things need not be always as they seem at the first glimpse..

This is such an example.

But still u have ended up in the same boat as me

coz of d reason we already discussed

some symmetric cancellation

As u see such symmetric cancellation always need not work.

Also im not sure abt ma method

bt this is a little better i feel!

maybe some1 will check over my work.

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