More Embarrassing Questions
For a statistical study, you need to figure out what proportion of people have a certain private attribute. But the attribute being private, no body really wants to tell you.
So, you devise the following protocol:
- You ask the subject to toss a fair coin. They do not show you the result of the toss.
- On Heads, they tell you their truthful answer.
-
On Tails, they toss the coin again and
- On a Heads, they tell Yes
- On a second Tails, they tell No
If in a sample of 100 people, 30 people tell you Yes , then what is the expected number of people who really have the private attribute?
The answer is 10.
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1 solution
Let probability of people to have it = x No of people with it = 100x
Case 1) Head in first toss (P=1/2) Subcase 1) Has it (P=1/2 * x => Number=100*x/2) Subcase 2) Does not have it (don't even care!)
Case 2) Tail in first toss (P=1/2) Subcase 1) Gets Head and tells yes(P=1/4 => Number =100/4=25) Subcase 2) Tails again and tells no ( don't even care!)
Therefore, Total yesses= 30 100*x/2 + 25 = 30 Solve for 100x 100x=10, Which is the number of people with the attribute. One doubt, Agnishom Dada, do you have a private attribute?