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Games of probability

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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?

Assume the die is fair

- Roll two dice.

Activity 2: Independence of two trials:

- Roll a die and toss a coin:
- What is the probability of getting a “3 and Tail” ?

- 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

- 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 ?

- What if the coin is not fair?

Answer: 1/6 * 1/3 = 1/18

Activity 3: Who is the winner?

- Toss a coin. Each time it’s head, you win $1, each time it’s tail, you lose $1.
- Roll two dice. Each time it’s 7, you win $4, otherwise you lose $1.
- Roll two dice. Each time it’s 7, you win $5, otherwise you lose $1.
- Roll two dice. Each time it’s 7, you win $6, otherwise you lose $1.

Even

Loser

Even

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.