Monty Hall

1

Monty Hall
This is a old problem, but it illustrates the
concept of conditional probability beautifully.
References to this problem have been made in much
popular culture, and a quick search on the
Internet will reveal much information.
Set this game up with 2 people, and three cups
and a prize hidden under the cups. Get one person
to act as Monty. Monty knows where the prize is
remember. Play a set of games where you always
stick and a set where you always change.
On a game show you are asked to choose one of
three closed doors. Behind one is a car, behind
another is a goat, and the other has nothing
behind it.
Monty will open a door after you have picked, and
then ask you if you want to stay with your choice
or switch to the other unopened door. Of course
he will never open the door you chose first.
What do you do - stick or change? Why?
What do you do - stick or change?
2

Constructing a tree diagram
Construct a tree diagram to show this information.
Tuesday
Rain
Monday
Rain
No rain
Rain
No rain
Keep the denominators the same in the final
working.
No rain
3

Probabilities
Calculate the probability that, a) it rains at
least once, b) it rains one day only, c) it
rains on one day only, given it rains at least
once.
It is possible to ignore the denominators here
only if they are the same at the end of your tree
diagram.
4

Questions
A teacher oversleeps with a probability of 0.3.
If he oversleeps then the probability of him
eating his breakfast is 0.2, otherwise it will be
0.6.
Misses breakfast
a)
0.8
0.24
Oversleeps
0.3
0.2
Eats breakfast
0.06
Misses breakfast
0.4
0.28
0.7
Does not oversleep
a) Construct a tree diagram to show this
information.
Eats breakfast
0.6
0.42
Use your diagram to find the probability
that, b) he oversleeps and does not eat
breakfast, c) he does not have breakfast, d) he
overslept, given he has breakfast, e) he
overslept, given he does not have breakfast.
0.24
b)
c)
0.52
d)
e)
5

Using tables
Conditional probabilities can be found simply
from data in tables, as illustrated by the
following. The table opposite shows the choices
of language and the gender of the 200 students
choosing those languages.
French German Total
Male 40 40 80
Female 90 30 120
Total 130 70 200
d) being female, given he/she does French.
A student is choosing at random, find the
probability of that student,
a) doing French,
e) doing German, given that he is male.
b) being male,
c) being male and doing German,