Probability progression, 6th step
The purpose of the activity
In this activity, the learners explore issues related to gambling to further develop their concept of chance. They calculate theoretical probabilities of game outcomes and develop an understanding of expected value.
The teaching points
- The probability of an event is a measure of the chance of an event occurring. The probability of an event is a number between 0 and 1 and can be expressed as a fraction, a percentage (0–100%), or as odds.
- There are two ways to measure chance. One way is to analyse the situation logically (theoretical probability), and the other way is to generate data to analyse the situation (experimental probability). Examples of situations that can be analysed theoretically include rolling dice, throwing coins and Lotto. Examples of situations that need to be analysed experimentally include the likelihood of there being an earthquake or a car accident.
- The expected value of an event is the sum of the probability of each possible outcome of the event multiplied by the value (or payoff) of the outcome. The expected value represents the average amount one ‘expects’ as the outcome of the chance event. A common application of expected value is in gambling. For example, a roulette wheel has 38 equally likely outcomes. A winning bet placed on a single number pays 35 to 1 (this means that you are paid 35 times your bet, and your bet is returned, so you get 36 times your bet). So, considering all 38 possible outcomes, the expected value of the profit resulting from a $1 bet on a single number is: ($35 x 1/38) + (–$1 x 37/38), which is about –$0.05. This means that the expected value of a $1 bet is $0.95 or that you lose, on average, 5 cents for every dollar bet.
- Discuss with the learners where expected value is used to determine values: insurance premiums, gambling pay-offs, sports betting.
- Cubes with 3 red faces, 2 green faces, and a blue face.
The guided teaching and learning sequence
1. Show the learners the cube with the coloured faces and write the following game rules on the board.
- The dice is rolled.
- You pay $2 to play.
- If the face is red, you lose the $2 it cost you to play.
- If the face is blue, you are paid $5 (you win $3).
- If the face is green, you are paid $3 (you win $1).
Tell the learners that they will start with $10 and have to stop playing if they run out of money. Play 10 rounds of the game with the class. Ask a volunteer to keep a running tally of the money won or lost.
2. Have the learners work in pairs and give each pair a cube with the same coloured faces as the previous cube and ask them to play a further 10 rounds of the game, keeping a tally of the money won and lost.
3. As a class, share the final outcomes of the games played.
“How many were winners? How many losers?”
“What was the largest winning total?”
“What was the largest loss?”
“Do you think this is a fair game to play?”
“Why or why not?”
“What is a fair game?” “How can you work out whether the game is fair or not?”
(A fair game is one where there are equal chances of wining and losing. In this case, it is a fair game if the cost of playing is equal to the winnings.)
4. Discuss with the learners the concept of ‘expected value’. Explain that the expected value of a probability situation is the sum of the probability of each possible outcome of the event multiplied by the value (or pay-off) of the outcome.
“What do you think the expected value of the cube game will be?” (Encourage the learners to link their answers to the class outcomes of playing the game.)
6. As a class, work out the expected value of the cube game:
“What are the possible outcomes in the coloured-cube game?” (blue, green, red)
“What are the chances of each of these happening?” (blue 1/6, green 1/3, red 1/2)
“What is the chance of winning $5?” (1/6)
“What is the chance of winning $3?” (1/3)
“What is the chance of losing $2?” (1/2)
“What is the expected value of playing the game?” (1/2 (–2) + 1/3 (1) + 1/6 (3) = –1 + 1/3 + 1/2 = –1/6)
“Is the game fair?” (no, in the end you lose 16.6 cents each turn you play.)
7. Ask the learners to work in pairs to invent similar dice games. Circulate as the learners are creating their games, asking them to explain if their games are fair or unfair.
8. Ask the learners to look at the games developed by other pairs. The challenge is to work out which of the games has the greatest expected value and which has the least.
Ask the learners to investigate whether the following game is fair. Two friends play a game where a single dice is rolled. Here is what happens for each result:
- Player 1 wins $3.
- Nobody wins or loses.
- Player 2 wins $5.
- Player 1 wins $3.
- Player 2 wins $4.
- Player 1 wins $2.
“Is it a good idea to play the game?”
“If this game isn’t fair, how could you change it to make it fair?
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