## Tuesday, September 15, 2009

### Probability of selling a house per viewing

The estate agent told us that a rule of thumb is eight viewings before your house is sold. But in these troubled times, it's been more like 16.

Gulp! So that got me thinking. What's the probability that someone viewing your house will actually buy it?

Looking for a simpler model, this is like throwing a coin over and over again until you get the first head (a throw is a viewing and a head is a sale). In this simple example, the probability of a throw resulting in a head is one half and the unknown is how many throws on average before you get the first head and can stop.

Expected number of throws before getting the first head:

1 throw = H: probability = 1/2 * 1 = 1/2
2 throws = TH: probability = 1/4 * 2 = 2/4
3 throws = TTH: probability = 1/8 * 3 = 3/8
4 throws = TTTH: probability = 1/16 * 4 = 4/16
...

You can see the infinite series this is going to create. I couldn't see any easy way to solve it*, so I messed around integrating x2-x from zero to infinity by parts (answer = 2.1) until I decided to check the Internet.

"Dr Anthony" has this cute recursive argument, I quote:

Let a = expected number of throws to first head.

We must make 1 throw at least and we have probability 1/2 of a head and probability 1/2 of returning to a, so

a = (1/2)1 + (1/2)(1 + a)

(1/2)a = 1

a = 2.

There you are: the expected number of throws before you get a head is just two.

So now we can adapt this argument for the estate agent. Let a be the number of viewings before the house gets sold (16 apparently) and let p be the probability of selling your house at each viewing. Then, following Dr Anthony,

a = p + (1-p)(1 + a)

So a = p + 1 + a - p - pa

and pa = 1, or p = 1/a.

What a simple result. Sounds like something I should have recalled.

So the chances of selling our house in 15 minutes time, when our first viewers arrive, is 1/16 = approx 6%. Wish us luck!
___________

Solution.

1. Write a(x) = 1/2 + 2x/4 + 3x2/8 + ... + nx(n-1)/2n + ...

2. Integrate to get a geometric progression with ratio (x/2).

3. Sum it to get x/(2-x).

4. Differentiate to get 2/(2-x)2

5. Set x=1 to get a(1) = 2.

This is the pedestrian way to match Dr Anthony's insight.