# Segment 3. Monty Hall

## Contents

#### Watch this segment

(Don't worry, what you see out-of-focus below is not the beginning of the segment. Press the play button to start at the beginning and in-focus.)

{{#widget:Iframe |url=http://www.youtube.com/v/Rxb8JG8nUFA&hd=1 |width=800 |height=625 |border=0 }}

The direct YouTube link is http://youtu.be/Rxb8JG8nUFA

Links to the slides: PDF file or PowerPoint file

#### Bill's Comments

You might enjoy reading some of the correspondence that Marilyn vos Savant received, on her web site.

### Problems

#### To Calculate

1. The slides used a symmetry argument ("relabeling") to simplify the calculation. Redo the calculation without any such relabeling. Assume that the doors have big numbers "1", "2", and "3" nailed onto them, and consider all possibilities. Do you still have to make an assumption about Monty's preferences (where the slide assumed 1/2)?

#### To Think About

1. Lawyers are supposed to be able to argue either side of a case. What is the best argument that you can make that switching doors can't possibly make any difference? In other words, how cleverly can you hide some wrong assumption?

2. We stated the problem as *requiring* the host to offer the contestant a chance to switch. But what if the host can offer that chance, or not, as he sees fit? Then, when offered the chance, should you still switch? (Spoiler alert: see this New York Times interview with Monte Hall.)

3. Mr. and Mrs. Smith tell you that they have two children, one of whom is a girl.

(a) What is the probability that the other child is a girl?

Mr. Smith then shows you a photo of his children on his iPhone. One is clearly a girl, but the other one's face is hidden behind the family dog, and you can't tell their gender.

(b) What is the probability that the hidden child is a girl?

(c) If your answers to (a) and (b) are different, explain why there is a difference.