The Monty Hall ProblemSuppose the contestants of a game show are given the choice of three doors: Behind one door is a car; behind the others, goats. After a contestant picks a door, the host, who know what is behind all the doors, opens one of the unchosen doors, which reveals a goat. He then says to the contestant, “Do you want to switch to the other unopened door?” Is it to the contestant’s advantage to make the switch?Choose door (1sttime)Door where the car is Choose door (2ndtime)AAStayABSwitchACSwitchBASwitchBBStayBCSwitchCASwitchCBSwitchCCStay

Three statements defining probabilitySuppose is a sample space associated with a random process A probability is a numerically valued function that assigns a number to every event such that
1)
2)
3)
If
is a sequence of mutually exclusive events, then
S
E
)
(
E
P
1
0
E
P
1
S
P
,...
,
2
1
E
E
1
1
i
i
i
i
E
P
E
P

Results following definition of probability
•
Result 1:
If events
and
are such that
then
•
Result 2:
The probability of an empty set is 0
1
E
2
E
2
1
E
E
2
1
E
P
E
P

Probability from a sample space with equally likely outcomes•Two events are said to be equally likely if one does not occur more often than the other EXAMPLEa) Consider the experiment of tossing a balanced die. Does the number 1 appear more often than the number 5?b) Consider the experiment of tossing three balanced dice. Does the number 10 appear more often than the number 9?