Case Singapore TOTO - case study for probability PDF

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DAO1704 Case: Singapore TOTO Game Design Top Toto prize snowballs to $6.5 million

The top Toto prize will snowball to about $6.5 million on Monday, as there has been no top prize winners for the last three consecutive draws. Nobody won the top Toto prize for the last week of last year, of which the prize money was almost $1.2 million. Nor did anyone win the top prize for Jan 4 and Jan 7, and winnings increased to over $4.7 million, reported Shin Min Daily News yesterday.

- See more at: http://news.asiaone.com/news/singapore/top-toto-prize-snowballs-65million#sthash.IxarwNal.dpuf

Background .

Prior to 7th October 2014, to play the game Toto, a player has to choose six numbers, ranging

from 1 to 45, and match them against the winning numbers drawn. In each draw, six winning numbers plus, one additional number will be drawn. The game is played on every Monday and Thursday in Singapore. If the numbers chosen match at least three numbers and the additional number drawn, the player wins a prize. For instance, if the player picked numbers 2, 7, 9, 14, 23 and 30, and the winning numbers drawn are: 2, 9, 14, 23, 30, 41 plus additional number 7, the player will win a prize for the 5+1 category (for matching 5 out of 6 winning numbers, and the additional number drawn).

Starting from 7th October 2014, pool of numbers from which the six-number combination are selected for the lottery draw will increase from 45 numbers to 49 numbers. Though it may be harder to strike the jackpot, punters may be happy to know that the prize money for the group 1 prize will be increased to a minimum of S$1 million from the previous minimum of $500,000. Currently the odds of striking the jackpot at the group 1 prize is at one in 8.1 million, while with the new rules, the chances of that have dwindled down to one in 13.9 million.

Singapore Pools is revamping the Toto game for the first time since 1997. Punters will find it more expensive to place bets now, with the minimum betting amount being increased from 50 cents to $1 for an ordinary bet on six numbers. This increase will also be reflected in the other system bets. For example, system 7 which allows the selection of 7 numbers is now increased to $7 from the previous $3.50. The consolation for many recreational punters is that there will be a new prize category, for a winning combination of three numbers. The prize money for this prize group is fixed at $10 per winning bet. The winners will get a share of the prize pool for each draw (“Prize Pool”). The Prize Pool shall be an amount equal to fifty-four per cent (54%) of the net amount of total stake collection of that draw less the Goods and Services Tax thereon1.

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We ignore the impact of GST in this case study

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While Singapore Pools will increase the winnings for the fifth and sixth prizes, it will reduce the prize money for the second, third and fourth prizes from 13% of the prize pool, to a declining prize allocation of 8%, 5.5% and 3%.

The total prize pool for the top 4 prize groups will be reduced from 72% of prize pool (in the 6/45 game) to 54.5 % in the new game. This reduction is to cater for the new prize group for the 3 number combination which will see to a large number of winners. The prize monies for the top 4 prize groups are pari-mutuel games, i.e., the prize monies are shared evenly among the winners. On the other hand, the prize monies for the last 3 groups are fixed-odds games, i.e., the prizes are given out for each $1 winning bet.

If there is no winner in Group 1, 2, 3 or 4, the prize amount will snowball to the next Draw, and added to the prize amount within the same group of the draw. After three consecutive draws without a winner for the Group 1 prize, the final jackpot amount will cascade in the fourth consecutive draw, and paid to the next prize group with winner(s) and shared equally.

Singapore Pools is a state-owned lottery subsidiary company in Singapore. As a wholly owned subsidiary of the Tote Board, it is the only operator that is legally allowed to run lotteries in Singapore. The proceeds from Toto betting, like the surpluses from other Singapore Pools games such as the Singapore Sweep and 4-D, are donated to causes such as the Community Chest.

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Exercise It can be seen from the sales data that a good majority of the 6/45 games in the last 3 years ended in the first (regular) draw. The sales volume of more than $3.5M in the regular draws translates to around 7M combinations played, which is close to the total number of combinations in the 6/45 game (8.14M number combinations). This resulted in frequent winners in the regular draws. In fact, only two draws ended in the 4th draw (no winner for JP for three consecutive draws) over the last 3 years.

Figure 1: Draw Number Distribution

Note that the risk of not generating enough reserve funds to maintain the minimum guarantee ($500K) is minimal in the current game, because even the regular draws are already able to attract routinely more sales to guarantee the minimum payment of $500K, often in the range $600K to $650K.

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Table 1: Sales (excluding special $5M and $10M draws)

Consecutive Draw No.

Average Average SALES

33% of Prize

1

3953275.703

704468.6364

2

5649559.411

1139073.123

3

9322621.889

1661290.833

Total

4688198.843

867631.7267

On the other hand, it is clear that snowballing have substantial impact on sales. Empirical evidence shows that sales increases at a faster rate as the draw number increases. The purpose of this exercise is to address the following issues: 1. The frequency of roll-over in the current 6/45 game is too low, as ticket sales has risen to produce more frequent winners of the Group 1 Jack Pot. Can the new 6/49 game attract more sales and lead to higher revenue for Singapore Pools? To address this question, draw the equivalent of Figure 1 and Table 1 for the new 6/49 game, based on the data available in http://www.singaporepools.com.sg/en/toto/Pages/results.aspx Estimate the sales revenue increase of the new 6/49 game for Singapore Pools. 2. The odds of winning any prize in the new 6/49 game should be sufficiently high to retain more effectively the interest of the players. What are the odds of winning a. Group 1 prize? b. Group 2 prize? c. Some monies in the 6/49 game?

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Present your analysis of the above questions using the information on Singapore Pools web site on sales revenue. See http://www.singaporepools.com.sg/en/toto/Pages/results.aspx

Hint: Based on the payout to the Group 1 prize, you can estimate the sales generated for each game fairly accurately. Use this information to estimate the impact of the change in game design on sales revenue.

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