Transcript Constructing a tree diagram

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,
On a game show you are asked
and three cups and a prize hidden
to choose one of three closed
under the cups. Get one person to
doors. Behind one is a car,
act as Monty. Monty knows where
behind another is a goat, and
the prize is remember.
the other has nothing behind it.
Play a set of games where you
Monty will open a door after you
always stick and a set where you
have picked, and then ask you if
always change.
you want to stay with your
choice or switch to the other
unopened door. Of course he will What do you do - stick or
change? Why?
never open the door you chose
first.
What do you do - stick or
change?
Constructing a tree diagram
3
The probability of rain in Manchester on any Monday is . If it rains
5
4
on a Monday, then the probability of rain on Tuesday is . If it does
5
3
not rain on Monday the probability of rain on Tuesday is .
5
Construct a tree diagram to show this information.
3 4 12
Tuesday
Rain
´ =
4
5 5 25
Monday
5
Rain
1
3
3 1 3
5
5
No rain ´ =
2
5
No rain
Keep the denominators
the same in the final
working.
5
3
5
Rain
2
5
No rain
5
25
2 3 6
´ =
5 5 25
2 2 4
´ =
5 5 25
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.
21
25
9
25
rain on one day only
rained at least once
9
9
=
9+12 21
It is possible to ignore the denominators here only if they are the same at
the end of your tree diagram.
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.
a)
0.3
0.7
a) Construct a tree diagram to show
this information.
Oversleeps
Does not
oversleep
b) he oversleeps and does not eat
breakfast,
b)
0.24
c) he does not have breakfast,
c)
0.52
d)
6
48
e)
24
52
e) he overslept, given he does not
have breakfast.
0.2
0.4
0.6
Use your diagram to find the
probability that,
d) he overslept, given he has
breakfast,
Misses
breakfast
0.8
Eats
breakfast
Misses
breakfast
Eats
breakfast
0.24
0.06
0.28
0.42
Using tables
Conditional probabilities can be
found simply from data in
tables, as illustrated by the
following.
Male
The table opposite shows the
Female
choices of language and the
gender of the 200 students
Total
choosing those languages.
A student is choosing at
random, find the probability of
that student,
a) doing French,
b) being male,
c) being male and
doing German,
130
200
80
200
40
200
French German
Total
40
40
80
90
30
120
130
70
200
d) being female, given
he/she does French.
e) doing German,
given that he is male.
90
130
40
80