GEM2900 CA Question

The following question is modified from the AY 09/10 Semester 1 CA3. I posted this as one of the few common kind of question that you are expected to solve for this module.

A marble is placed in the bag. The marble is equally likely to be black or white and you don't know its colour. Next, another marble is placed in the bag, this time it's a white marble (you do know the colour). The bag is shaken and one of the marbles is drawn at random. It is white. What is the probability that the marble remaining in the bag is also white? Now, we put it back white marble into the bag and perform the another draw. What is the chances that we will see a white marble on the second draw?

Risk Taking - League Table Quiz

A mini challenge for soccer punters

(Disclaimer: I am not promoting gambling! But if you really choose to gamble, do it responsibly!)

I chanced upon this question while I was browsing through books on psychometric test. It was mentioned in the book that this is the hardest question that can possibly appear in the numerical reasoning test. However, I strongly believe that any punters that place bets for soccer games should be able to solve this question.

Suppose there are 6 soccer teams: Team Alpha, Team Beta, Team Delta, Team Gamma, Team Omega and Team Sigma. As usual, a team will receive 3 points if it wins a match but will not receive any point if it loses that match. However, if the game ends in draw, both teams will each receive 1 point for that game. Let us also assume that each team will play with the remaining 5 teams once for the whole season. The following is the league table for that 6 soccer teams:

                                      No of matches played          Total Points
Team Alpha                                   4                                   10
Team Beta                                     4                                    9
Team Delta                                    4                                    6
Team Gamma                                4                                    6
Team Omega                                 4                                    1
Team Sigma                                   4                                    0

Q1) So far, how many matches have taken place?

Q2) Which of the above-mentioned team(s) can still be the champion for this season?

Q3) Which of the above-mentioned team(s) can still finish last for this season?

Risk Taking - European Odds

In this article, I will teach you how to interpret a set of European odds.

(Disclaimer: I am not promoting gambling! But if you really choose to gamble, do it responsibly!)

For learning purpose, I shall use a coin-flipping game. There are only one bookmaker and one punter in this example. The coin used is believed to be fair and hence the chance of showing a head is the same as the chance of showing a tail. Lastly, we also assume that the bookmaker is not out to make money or lose money after many rounds of coin-flipping game.

The most rational set of odds set by the bookmaker will be 2 for Head and 2 for Tail. It means that if punter bet Head and the coin indeed shows Head, the punter will win $1 for every $1 the punter bets. However, if the coin shows Tail, the punter will lose every dollar that the punter bets. Why is the odd set at 2? It is because the probability of showing Head for a fair coin is ½. Over a long run, the bookmaker has 50% chance of losing the bet and 50% chance of winning the bet. If the bookmaker loses the bet, the bookmaker will lose $1 for each $1 bet placed by the punter. If the bookmaker wins the bet, the bookmaker will win $1 for each $1 bet placed by the punter. Over the long run, the expected gain by the bookmaker = ½(-$1)+1/2(+$1) = $0 for every dollar betted. This is what the bookmaker wants – not making or losing money after many games.

In fact, if the odds for Head = 1/x and the odds for Tail = 1/y, it means that bookmaker BELIEVES that the chances of the coin getting Head is x and the chance for Tail is y.

So, in what situation should a rational person enter the game and place the bet?

A smart punter will enter a game if the punter does not agree with the probabilities that the bookmakers believe. If punter somehow manage to get the coin to show Tail all the time (without alerting the bookmaker), it means that the punter believes that the probability of Tail =1. As the foolish bookmaker still thinks that the coin is fair, he/she will still believe that probability of Tail =1/2 (despite using an unfair coin). The smart punter will definitely enter the game and bet on Tail. Over the long run, the expected return of the punter is 1(+$1)+0(-$1) = $1 for every dollar betted. The smart punter is better off by betting. However, the smart punter will stay out from betting Head as probability of Head perceived by bookmarker > probability of Head perceived by the smart punter. The punter will be worse off betting Head as the expected return will be 0(+$1)+1(-$1) = -$1 per dollar betted.

In the next article, I will discuss about bookmaker’s rule and the non-existence of arbitrage in a single betting game if the bookmaker is rational.

Risk Taking - Odds

In this article, I seek to provide an intuitive understanding to the idea of odds.

(Disclaimer: I am not promoting gambling! But if you really choose to gamble, do it responsibly!)

I will be using the decimal odds system in this article. This is the odds system used by Singapore Pools (the only legal bookmaker in Singapore). ‘Decimal odds, also known as European odds, express the amount of money which will be returned to the punter on a 1 unit stake. For example, if Barcelona is favoured to win at 1.50, then a $100 bet will give you back $150 in total. ’ <http://www.online-betting.me.uk/articles/how-to-read-betting-odds.html> If Barcelona really wins the match, the punter will get back your $100 capital and enjoy a winning of $50. However, if Barcelona loses the match, the punter will lose the $100 capital.

1) How does the bookmaker come out with odds?

I am not sure. Most probably, the bookmaker uses some form of complicated statistical model.

2) What information can punter obtain from the odds provided by the bookmaker?

We can know about bookmaker’s view for that game. If the odds are lower as compared to other options (such North Korea winning that match), it means that the bookmaker thinks that Barcelona has a high chance of winning the match. However if Barcelona's odds are higher as compared to other options, it means that the bookmaker thinks that Barcelona has a lower chance of winning the match.

3) Should you place a bet for an option with higher or lower odds?

It is really up to you. For an option with higher odds, the chance of occurring is lower but the reward is higher. For an option with lower odds, the chance of occurring is higher but the reward is higher. It is risk-reward issue.

In the next article, I shall be using some simple probability ideas to discuss more interesting issues such as, when should you (a rational person) place bet for a game?