
The
Birthday problem Surprise  Surprise !
A challenge to your intutive
sense but quite simple in a mathematical sense!
(A) Let's start with an easy one first.
At
least how many people must be gathered so that we can be 100% certain that
some of them share the same birthday?
(B)
Now comes the serious part ...
How
many people should there be in a class so that there is at least a 50%
chance that some of them share the same birthday?
Should
it be half of what we have in (A) above? How about one third? say 180
people? 120 people? Think about your own class ( around 40 in size ),
do you know any 2 people having the same birthday? Ask another class,
is it common to have 2 or more people having the same birthday?
Discussion
(1)
A
computer simulation
(2) How
to approach the problem mathematically? (Assume there are only 365 different
birthdays.)
(a) 
Suppose
there are 3
people in a class. Find the probability that 

(i) all
have different probability (ii) at least some have same birthdays

(b) 
Repeat
(a) with 5 people
in a class. How about n
people? Guess how large should n
be so that the probability that some of them have the same birthday is
greater than 0.5. 
(3) A full explanation
Does the answer agree with your intuition?


The famous
Monty
Hall Problem *
Suppose
you're on a game show, and you're given the choice of three doors.
Behind one door is a car, behind the others, goats. You pick a door,
say number 1, and the host, who knows what's behind the doors, opens
another door, say number 3, which has a goat. He says to you, "Do you
want to pick door number 2?" Is it to your advantage to switch your
choice of doors?
If you were the contestant, which of the following
would have a better chance to win the big prize?
Strategy 1 (stick): Stick with the original door
Strategy 2 (switch): Switch to the other door
or it doesn't matter since the two strategies have
equal chance of winning the big prize
*This
problem was named Monty Hall in honor of the long time host of the
American television game show "Let's Make a Deal." During the show,
contestants are shown three closed doors. One of the doors has a big
prize behind it, and the other two have junk behind them. The
contestants are asked to pick a door, which remains closed to them.
Then the game show host, Monty, opens one of the other two doors and
reveals the contents to the contestant. Monty always chooses a door
with a gag gift behind it. The contestants are then given the option to
stick with their original choice or to switch to the other unopened
door.
Discussion
(1) 
A
computer simulation for the Monty Hall problem
 "Let's make a deal" applet 

just scroll down the
page until you see 3 large doors #1, #2 and #3. Try the simulation
repeatedly with your friends : one using the always STICK door strategy
and the other using the always
SWITCH door strategy. Do the statistics shown in
small print below the 3 doors deviate significantly between the 2
strategies? Is there a clear winner and does that agree with your
intuition? 


(2) 
The winning strategy is ...
with FULL explanation below 

There
is a 1/3 chance that you'll hit the prize door, and a 2/3 chance that
you'll miss the prize. If you do not switch,
1/3 is your probability to get the prize. However, if you missed (and
this with the probability of 2/3) then the prize is behind one of the
remaining two doors. Furthermore, of these two, the host will open the
empty one, leaving the prize door closed. Therefore, if you miss and then switch, you
are certain to get the prize! Summing up, if you do not switch your
chance of winning is 1/3 whereas if you do switch your chance of
winning is 2/3! More discussion 


(3) 
History
of the Monty Hall problem 

This problem has aroused a heated debate for
quite some time when it first appeared in 1991 and has never failed to
vex and amaze people with its counterintuitive solution ever since.
In 1991, Marylin Vos Savant** received the
Monty Hall problem from Craig. F. Whitaker (Columbia, MD). She
carefully explained the logic of the correct solution in a number of
subsequent columns, but never completely convinced the doubters.
Marylin's response caused an avalanche of correspondence, mostly from
people who would not accept her solution (particularly advocates of
"5050" school). Eventually, she issued a call to Math teachers among
her readers to organize experiments and send her the charts. Some
readers with access to computers ran computer simulations. At long
last, the truth was established and accepted. The matter continued at
such length that it eventually became a notable news story in the New
York Times and elsewhere.
**Marylin Vos Savant ran
the popular "Ask Marylin" questionandanswer column of the U.S. Parade
magazine. According to Parade, Marilyn vos Savant was listed in the
"Guinness Book of World Records Hall of Fame" for "Highest IQ" with IQ
score of 228.

