Gambling and probability 1. Odds and football

 Predict the Premier League results for this weekend.  Can you estimate the probability of a win/draw/loss for each of the 10 home teams? Place your bets 2

Converting betting odds to probability. If a horse has odds of 10 to 1 to win, what is the probability of that horse winning? Betting odds 3

Odds represent the chance or likelihood of something happening Odds are based on probability What are odds? 4

The odds in favour of an event occurring are: Odds in favour p = probability of event occurring 1 – p = probability of event not occurring ➞ p : 1 – p 5

The odds against an event occurring are: Odds against p = probability of event occurring 1 – p = probability of event not occurring 1 – p : p ➞ 6

What are the odds that you will randomly guess the day of the week correctly? 27 July 2073  This ratio simply means that it is six times more likely to be a day other than Sunday (which is a 1 in 7 chance).  This is why odds of 1:1 are know as even odds, because you have a 1 in 2 or even chance of winning. 1:6 Why does this not mean a 1 in 6 chance? 7

Odds in gambling are always presented as odds against and not for Odds in gambling e.g. 4:1 The probability of losing is four in five 8

Winnings The house pays £10 for every £1 bet, plus you get your original stake back You will make £10 profit for every £1 bet £11 If you gamble £1 on odds of 10:1 and win, what do you think you will get back? 9

 Whilst odds are always in their simplest form, they must consist of whole numbers.  Which is why not all odds are ‘something to one’, e.g. 7 : 2  All this means is that for every £2 your profit is £7 Something to One  7 : 2 is the same as 3.5 : 1 £3.50 If you bet £1, what would your profit be? 10

 What is the assumed probability of success with the following odds? –7:1 –5:1 –9:4  If you bet £5 on each of the above odds, calculate your: – Total winnings – Profit Questions 11

 What is the assumed probability of success with the following odds? –7:1 –5:1 –9:4  If you bet £5 on each of the above odds, calculate your: – Total winnings – Profit Solutions one eighth one sixth four thirteenths (£40, £30, £16.25) (£35, £25, £11.25) 12

What are the possible outcomes in a football game?  Win  Draw  Loss  What should the probability add up to?  What does it add up to?  Convert the betting odds for all 10 league games this weekend, then convert them to probabilities.  What does the probability of win/draw/loss add up to in each game? Why does the bookie always win? 13

 Covert the odds to probabilities.  What are the possible outcomes for all three matches?  What is the probability of each outcome?  Can you represent this on a tree diagram?  If I bet £5 on each outcome, which outcome would get me the most money? Extension: Take all the upcoming matches this weekend; which three outcomes would gain you the most money if you bet £5? Take three teams from the Premier League and look at their probabilities for a win/draw/loss this weekend. 14

What is an accumulator bet?  An accumulator is a bet that combines four or more selections into a single wager that gains a return only when all parts win.  The advantage of an accumulator is that winnings are much higher at the expense of increased risk, but only a single selection needs to lose for the entire bet to lose.  An accumulator is a bet of four or more selections; those with fewer than four behave the same but have different names, e.g. double or treble. Football accumulators 15

Based on the predictions you made at the start of the lesson, what is the probability that all 10 results are correct?  If you bet £1, how much would you win? Create a spreadsheet with the probabilities of all 10 games. Then answer the following questions:  What is the least likely set of outcomes?  What is the most likely set of outcomes?  What is the probability all matches are a draw? Football accumulators 16

Example spreadsheet 17

Gambling and probability 2. Blackjack and Roulette

 Make 21 or have a higher score than the dealer  Anything over 21 you are bust  If you are equal to the dealer no one wins or loses  Picture cards are worth 10, Aces are 1 or 11  You are dealt one card each and then must place your bets  If you get 21 through a face and an ace you get your money and a half back. How to win in Blackjack 19

 What do you think the probability of getting 21 is?  How many combinations, from one pack of cards, can you make 21 from just picking two cards?  Try using a space diagram and tree diagrams to help you.  Extension: play with four packs of cards.  What happens if you can have more than two cards: do you think this will increase your probability? Making 21 20

 Why does the dealer have the advantage?  Based on the cards on the table, can you find the probability of getting the card you want?  Why would counting the cards be beneficial?  What strategy would you use? Play some Blackjack 21

 A roulette wheel has 36 numbered red and black slots, plus a green zero  Single numbers pay 35:1  Two adjoining numbers pay 17:1  Odd/even, black/red pay evens (1:1) Roulette 22

1.Find the probability and percentage of winning when betting on: 1; 2; 3; 4; 6; 12 and 18 numbers 2.Find the percentage difference from the real odds and the casino odds when betting on: one number; two numbers or 18 numbers. What do you notice? Does this change your odds when playing? 3.Find the percentage that the same colour will appear twice. How many times would the same colour have to appear before the odds became less than if you bet on just one number? 4.Find the percentage of the same number appearing twice in a row. If you bet £5 on this, what would you win and what should you have won? Give the percentage difference. Roulette 23

1.Find the percentage of: –Same number twice in a row –Landing on green –Green appearing within any 10 spins –The same colour over two spins –The colour and even/odd correctly –The colour and the dozen correctly –The dozen and the column correctly 2. Which of the above should you bet on and why? Extension 24

2.2.7% – No, it is the amount the casino keeps from your winnings %; 6 spins (see next slide) %; £360 – should be £370; 2.7% Roulette – answers Roulette bets probability chart 25

Same even money bet result in a row probability How unlikely is it to see the same colour two or more times in a row? What is the probability of the results of five spins of the roulette wheel being red? The chart here highlights the probabilities of the same colour appearing over a certain number of spins of the roulette wheel. Same colour in a row 26

2. Colour and even/odd as it has the best odds. 1. Random roulette probabilities: Extension – answers 27

 Pick two quarters of the board (they can be the same quarter, overlapping, opposite ends etc.).  We will ‘spin’ a number of times and you will record how many times you win and lose (profit or loss).  Remember: if 0 is landed on, the ‘Casino’ wins! Play Roulette! 28

Core Maths Support Programme 60 Queens Road Reading RG1 4BS Call