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Statistics and Truth
This was a nice concept I learned from the book Statistics and Truth by the "living Indian great" C R Rao.
Let's assume that we have to survey a bunch of people; We have all the details about them, like, for example, their cell phone nos (every one in the bunch has one). But, what we don't know is how many of them smoke marijuana (it is everyone's dark secret). So, we simple cannot survey by asking them this question directly (they will want to lie and escape the long hands of the law).
What i am going to explain is a mind blowing method to conduct such a survey.
1) Ask everyone to come to your room for the interview in order.
2) give them them following instruction: "See this coin in the table, flip it.
If head comes, answer the question: "does your phone no end in an even digit?"
If tail comes, answer the question: "do you smoke marijuana?"
3) Now the guys who came for the survey know that, their secret is maintained (You never know which question they are answering). So, they proceed boldly.
4) Person one flips the coin (he does not show you whether the outcome was heads or tails)... Says "yes"/"no" (without showing any facial expression). You note down his answer (with a grin) and say "next person". You proceed Ad nauseum for all the others in the list.
5) In the end, you have a list of noted down responses, "yes"s and "no"s.
6) Except that you don't know who answered which question.
7) the fun thing is: you don't have to. All you want to know is how many smoke marijuana... not their identities.
8) You forget about the "no"s.. they mean, the person who answered it either did not smoke marijuana or his phone no. does not end in a even no. In any case, this information is of no interest to us.
9) The "yes"es matter. Some of those "yes"es were answer for the question: "do you somke marijuana?" and the other "yes"es were answer for the question" "does your phone no end in an even no.?"
10) But you don't know how many (among the "yes"es) answered the first question and how many answered the second.
11) Now, keep this fact in mind. You know how many people's cell. no. ends with an even no. because, as i have pointed out at the start, you already have that information with you. Example, there were 100 people out of whom 46 have cell no's which end in even no.
12) Also keep this fact in mind, on an average, 50% of the tosses of a coin turns up to be heads and 50% turns out to be tails. (For this you must know probability. Both the outcomes are equally possible and so have 1/2 prob. each).
13) This is the hard part. Now, only consider those people who have their cell. nos ending with an even no.
Out of these people, 50% got heads and 50% got tails. i.e. out of those who have a cell. phone. ending with an even. no., only 50% get a "chance" to answer the first question. So, out of the 46 people only 23 got a "chance" to answer the first question. So, out of the no of "yes"es 23 of them were meant to answer the question "do you have a cell phone. ending with an even no?"
14) Now comes the nice part (given that u understood part no. 13), you know the no. of yeses (count them in the list).... say 62. 23 of them were meant to answer question no. 1. So, the other 39 "yes"es were the answer to the question "do you smoke marijuana?"..
15) Just like, only half of the people who have their cell phone nos got a "chance" to answer the first question. Only half of the people who smoke got a "chance" to answer the second question. i.e. Only "half" of the people who smoke marijuana, got a "chance" to say yes. So 39 "yes"es is only half of those who smoke marijuana.
16) So, how many DO smoke marijuana,. 39*2= 78.
17) Tadaaa.
Formally,
p=no. of yeses.
a=no. of people who have cell ph ending with an even no.
b=no. of people who smoke marijuana.
a/2+b/2=p
b/2=p-a/2
b=2p-a
tada.
Now, it is not true that a coin ALWAYS turns up heads 50% percent of the times. It is almost always close to 50% of the times. And, if the no. of tosses increases, the no. of heads, gets much closer to 50%.
If you are unconvinced by the above to lines of text, read probability.
Thanks to Rahul Gupta for persuading me to learn this concept.
Let's assume that we have to survey a bunch of people; We have all the details about them, like, for example, their cell phone nos (every one in the bunch has one). But, what we don't know is how many of them smoke marijuana (it is everyone's dark secret). So, we simple cannot survey by asking them this question directly (they will want to lie and escape the long hands of the law).
What i am going to explain is a mind blowing method to conduct such a survey.
1) Ask everyone to come to your room for the interview in order.
2) give them them following instruction: "See this coin in the table, flip it.
If head comes, answer the question: "does your phone no end in an even digit?"
If tail comes, answer the question: "do you smoke marijuana?"
3) Now the guys who came for the survey know that, their secret is maintained (You never know which question they are answering). So, they proceed boldly.
4) Person one flips the coin (he does not show you whether the outcome was heads or tails)... Says "yes"/"no" (without showing any facial expression). You note down his answer (with a grin) and say "next person". You proceed Ad nauseum for all the others in the list.
5) In the end, you have a list of noted down responses, "yes"s and "no"s.
6) Except that you don't know who answered which question.
7) the fun thing is: you don't have to. All you want to know is how many smoke marijuana... not their identities.
8) You forget about the "no"s.. they mean, the person who answered it either did not smoke marijuana or his phone no. does not end in a even no. In any case, this information is of no interest to us.
9) The "yes"es matter. Some of those "yes"es were answer for the question: "do you somke marijuana?" and the other "yes"es were answer for the question" "does your phone no end in an even no.?"
10) But you don't know how many (among the "yes"es) answered the first question and how many answered the second.
11) Now, keep this fact in mind. You know how many people's cell. no. ends with an even no. because, as i have pointed out at the start, you already have that information with you. Example, there were 100 people out of whom 46 have cell no's which end in even no.
12) Also keep this fact in mind, on an average, 50% of the tosses of a coin turns up to be heads and 50% turns out to be tails. (For this you must know probability. Both the outcomes are equally possible and so have 1/2 prob. each).
13) This is the hard part. Now, only consider those people who have their cell. nos ending with an even no.
Out of these people, 50% got heads and 50% got tails. i.e. out of those who have a cell. phone. ending with an even. no., only 50% get a "chance" to answer the first question. So, out of the 46 people only 23 got a "chance" to answer the first question. So, out of the no of "yes"es 23 of them were meant to answer the question "do you have a cell phone. ending with an even no?"
14) Now comes the nice part (given that u understood part no. 13), you know the no. of yeses (count them in the list).... say 62. 23 of them were meant to answer question no. 1. So, the other 39 "yes"es were the answer to the question "do you smoke marijuana?"..
15) Just like, only half of the people who have their cell phone nos got a "chance" to answer the first question. Only half of the people who smoke got a "chance" to answer the second question. i.e. Only "half" of the people who smoke marijuana, got a "chance" to say yes. So 39 "yes"es is only half of those who smoke marijuana.
16) So, how many DO smoke marijuana,. 39*2= 78.
17) Tadaaa.
Formally,
p=no. of yeses.
a=no. of people who have cell ph ending with an even no.
b=no. of people who smoke marijuana.
a/2+b/2=p
b/2=p-a/2
b=2p-a
tada.
Now, it is not true that a coin ALWAYS turns up heads 50% percent of the times. It is almost always close to 50% of the times. And, if the no. of tosses increases, the no. of heads, gets much closer to 50%.
If you are unconvinced by the above to lines of text, read probability.
Thanks to Rahul Gupta for persuading me to learn this concept.
