10/7/2009
Test 1
Introduction to Probability
Probability is Not Always Intuitive
• Mean: 52.44/75 = 69.92% • A: 67 – 75
• Median: 58/75 =77%
• B: 60 – 66
• Min: 3 Max: 69
• C: 51 – 59
• D: 40 – 50
• F: 0 – 39
3
16
11
7
8
"Let's Make a Deal"
• Ex.: The "Let's Make a Deal" Paradox
• "Let's Make a Deal" was the name of a popular television game
show in the 1970s. In the show, the contestant had to choose
between three doors. One of the doors had a big prize behind
it such as a car or a lot of cash, and the other two were empty.
• The contestant had to choose one of the three doors, but
instead of revealing the chosen door, the host revealed one of
the two unchosen doors to be empty. At this point of the
game, there were two unopened doors, one of which had the
prize behind it - the door that the contestant had originally
chosen, and the other unchosen door.
• The contestant was given the option to either stay with the
door they had initially chosen or switch to the other door.
What do you think the contestant should do, stay or switch?
• http://www.stat.sc.edu/~west/javahtml/LetsMakeaDeal.html
Explanation
• The intuition of most people is that each of the
two doors is equally likely to contain the prize so
there is a 50-50 chance of winning with either
selection. This, however, is not the case!
Actually, there is a 66% chance (or probability of
2/3) of winning by switching, and only a 33%
chance (or probability of 1/3) of winning by
staying with the door that was originally chosen.
This means that a contestant is twice as likely to
win if he/she switches to the unchosen door!
Isn't this a bit counter-intuitive and confusing? I
thought so when I was first faced with this
problem.
Birthday Example
Pick this door
Case I
Case II
A
B
C
0
0
X
$
0
$
Switch and win!
$
0
X
Switch and win!
0
X
0
Stay and win!
Case III
• Suppose that you are in a group
with 59 other people (for a total of
60). What are the chances (or,
what is the probability) that at
least two of the 60 people share
the same birthday?
1
10/7/2009
Birthdays
• The probability that at least two of the 60
party guests share a birthday is:
– Quite small—somewhere between 1% and
10%
– About 1/6≈17%--since there are 60 and
365 days (365/60 ≈1/6)
– About 50% chance
– Quite high—somewhere around 75%-85%
– Very high—above 99% chance
Birthdays
• Surprisingly, there is a 99.6%
chance that at least two of the 60
guests share the same birthday. In
other words, it is almost certain
that at least two of the guests
share the same birthday. Now this
is what I call counter-intuitive!
2