Games of probability
What are my chances?
Activity 1: Simple probability:
• Roll a single die (6 faces).
– What is the probability of each number showing on top?
Number on Probability
top
1
?
1/6
2
?
1/6
3
?
1/6
4
?
1/6
5
?
1/6
6
?
1/6
Assume the die is fair
• Roll two dice.
Sum of two dice
Probability
Sum of
two dice
Can be done by:
Probability
2
1/11 (?)
2
(1, 1)
1/36
3
3
(1, 2) (2, 1)
2/36
4
4
(1, 3) (2, 2) (3, 1)
3/36
5
5
(1, 4) (2, 3) (3, 2) (4, 1)
4/36
6
6
(1, 5) (2, 4) (3, 3) (4, 2) (5, 1)
5/36
7
7
(1, 6) (2, 5) (3, 4) (4, 3) (5, 2) (6, 1)
6/36
8
8
(2, 6) (3, 5) (4, 4) (5, 3) (6, 2)
5/36
9
9
(3, 6) (4, 5) (5, 4) (6, 3)
4/36
10
10
(4, 6) (5, 5) (6, 4)
3/36
11
11
(5, 6) (6, 5)
2/36
12
12
(6, 6)
1/36
Activity 2: Independence of two trials:
• Roll a die and toss a coin:
– What is the probability of getting a “3 and Tail” ?
die
Coin
Coin
Probability
1
H
1/12
1
T
1/12
2
H
1/12
2
T
1/12
3
H
1/12
3
T
?
1/12
4
H
1/12
4
T
1/12
5
H
1/12
5
T
1/12
6
H
1/12
6
T
1/12
• Probability of getting a 3 on the die
= 1/6.
• Probability of getting a tail on the
coin = 1/2
• Since the outcomes of the coin toss
and the die rolling are independent,
the join probability of getting a 3 AND
a tail is (1/6)*(1/2)=1/12
• What if the coin is not fair?
• Assume the odds of getting the tail from the coin is 1/3,
head is 2/3.
– What is the probability of getting a “3 and Tail” now ?
Answer: 1/6 * 1/3 = 1/18
die
Coin
Probability
1
H
2/18
1
T
1/18
2
H
2/18
2
T
1/18
3
H
2/18
3
T
1/18
4
H
2/18
4
T
1/18
5
H
2/18
5
T
1/18
6
H
2/18
6
T
1/18
Activity 3: Who is the winner?
1.
2.
3.
4.
Toss a coin. Each time it’s head, you win $1, each time
it’s tail, you lose $1.
Even
Roll two dice. Each time it’s 7, you win $4, otherwise
you lose $1.
Loser
Roll two dice. Each time it’s 7, you win $5, otherwise
you lose $1.
Even
Roll two dice. Each time it’s 7, you win $6, otherwise
you lose $1.
Winner
Activity 4: Don’t be fooled
•
3 piles of cards. 2 cards in each pile:
– Pile 1: ♥K and ♥K
– Pile 2: ♥K and ♠K
– Pile 3: ♠K and ♠K
•
We don’t know which pile is which. Randomly pick one
card from one pile. If the card we pick is ♥K, what is
the odds that the other card in the pile is also ♥K?
•
Let’s do an experiment!
• Ways to pick ♥K :
– if we happen to pick a card from pile 1: either card will do.
– If we happen to pick a card from pile 2: only one card will
do.
– If we happen to pick a card from pile 3: no card will do.
• Probability of picking ♥K :
(1/3)*(1)+(1/3)*(1/2)+(1/3)*0=1/2
• Probability of picking a pile which has two ♥K: 1/3
• So, knowing one card is ♥K, the probability of the other
one is also ♥K is (1/3)/(1/2)=2/3
• Will you be a winner if you play this game?
– Each time when ♠K is picked, no win, no lose.
– Each time when ♥K is picked, you win $3 if the other
card is ♠K.
– Each time when ♥K is picked, you lose $2 if the other
card is ♥K.
You win $3 when you win, you only lose $2 when you
lose…. Do you think you can make money by playing
on?
NO!
DON’T BE FOOLED!
• Most of the gambling games are like this
example – The odds are not in favor of the
player.
• Use the concept of probability can help
you determine whether a decision is good
or bad – such as making investments.
• Don’t gamble – unless your math tells you
that you can win.