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### Survey Respondents Flip Coins

```
Date: 08/04/97 at 07:21:17
From: Anonymous
Subject: Probability

I have an advanced math class and I'm stuck on a problem, which is:

Respondents are asked to flip a coin in private and answer question A
if the result is a head and question B if the result is a tail.
Knowing the probability of getting a head or a tail and the
probability that question B will be answered "Yes" (it's 0.5), the
researcher can then use the total fraction of "Yes" responses to
calculate the probability that question A will be answered "Yes."

If this survey is administered to 1000 people and 700 people answer
"Yes", find the probability that a person will answer "Yes" to
question A.
```

```
Date: 08/04/97 at 16:09:29
From: Doctor Anthony
Subject: Re: Probability

These problems are best done by laying out the probabilities in
tabular form:

|  A(1/2)          B(1/2) |
--------------------------------------
YES  |  .45              .25   |  .7
|                         |
|                         |
NO  |  .05              .25   |  .3
----------------------------------------
|  .50              .50   |   1

This table is completed by filling in  .25 (= 1/2 x 1/2) in the
B/YES box, then .7 as the total for the YES row, so this means
A/YES must be .45

B/NO must also be 1/2 x 1/2 = .25 so the B column is .5

The overall total is 1, so A column is also .5  This means that
A/NO must be .05   We have been able to complete the table by
noting the sums of rows or columns and one or two interior
probabilities.

The .45 in the A/YES box is the product of the probabilities:

Probability of getting question A x probability of answering this YES

(1/2) x (prob. of answering YES) = 0.45

Prob. of answering YES to question A = 0.90

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Probability

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