If you shop at Amazon anyway, consider using
this link. We receive a few cents from each purchase.
If you benefit from the website, in terms of
knowledge, entertainment value, or something otherwise useful,
consider making a donation via PayPal to help defray the
costs. (No PayPal account necessary to donate via credit
card.) Transaction is secure.
e-mail with your comments about this program (or anything else).
You have a choice of one of 3 doors. One of the doors has a
new car behind it, the other two have goats as prizes. After you select a door, the host will open one of the other
doors revealing a goat. You then have the choice of sticking with your original choice or switching doors.
Assuming that it is more desirable to take home a car than a goat, what should you do? Does it make any difference?
Background & Techniques
I ran across this as a math puzzle the other day, and got the answer
wrong. So as "punishment"
I forced myself to write a program illustrating the probability
Here are the program implementation highlights:
- A TDoor object was defined to keep track of
whether or not the door has been selected by the player, which prize is behind
the door, whether or not it is open, and a pointer to the TShape
rectangle that represents the door. If it were a real visual component,
it could have been a descendant of TShape, but this is much simpler for
casual applications - just set a pointer to the "door" Tshape
rectangle as a property at the time that we create each TDoor. The
form has a Doors array containing the 3 TDoor
- I had some fun animating the doors. I decided
to "raise" them vertically to reveal the hidden prize, a
whole lot easier than designing swinging doors. That part was easy
- just reduce the height property of a rectangular Tshape object (the door) in
a loop. Oh, and set form property doublebuffered to true to
stop that flicker problem.
- The harder part was getting the pictures of the
prizes rearranged behind each door. I wrote a Shuffle
routine that uses the idea commonly used to programmatically shuffle a deck of
cards. Load the cards numbers, picture numbers in this case, into an
array and then run though the array exchanging each entry with one
randomly selected from those above (or below) it. The effect is of
moving randomly selected cards from the un-shuffled portion of the deck to
the bottom (or top) of the deck. The resulting arrangement of picture numbers is used
to draw each picture behind the appropriate door. The
"behind" part is also easy. I loaded the pictures originally
so that they exactly overlapped the door images. A right click and
"Send to back" on each moved it to the top of the the Z-order
list. Windows draws screens by processing this list from top down,
so high stuff is drawn first and then can be covered by objects lower in the
Z-order list. So the doors automatically hide the prizes.
- The actual door selection is handled in a MouseDown
event exit. If no door has been selected yet, mark the clicked door as
selected and randomly find an unselected closed door hiding a goat for the
host to open. If a door has already been selected, then this is
the player's second click and we can complete the game. That mainly
means identifying if he switched doors and if he won or lost and adding the
results to some totals buckets.
- I did need to add a Busy flag to ignore extra clicks while processing a click.
- After the explanation dialog has been displayed, we
make visible a button that will play 1000 games at a time (without
animation). The button click code is pretty much much a copy of
the MouseDown exit code, except we simulate the clicks and skip the
You'll have to download and run the program to discover
the answer and explanation for the original question.
Running/Exploring the Program
Suggestions for Further Explorations
||The game was
originally posed to Marilyn Vos Savant in her weekly column in Parade
magazine. It prompted 10,000 response letters, most of telling
her that her answer was wrong (it wasn't). You can
easily spend an hour or so investigating the results from a Google
search on "car goats Marilyn".
February 18, 2016