Friday, January 8, 2016

Will there be a Powerball Winner on Saturday? - And, What is the probability there will be?

The Powerball jackpot for tomorrow night's drawing (Saturday January 9 2016) has just been increased to $800 million for the annuity value, and to $496 million for the cash value. Based on this, we are estimating that around 400 million tickets will be sold in total for the next drawing.

Since the total number of combinations is 292.2 million, one would think that the jackpot will be won for sure (because 400 is greater than 292).

But you would be wrong.(See Probability of Winning the Lottery)

The real probability that the winning ticket will be sold is only:

a 74.6% probability that Powerball will be won 
(if 400 million tickets are sold).

This means that there is a 25.4% chance that there will be no winner, and that the jackpot will continue to grow!

The explanation for estimating this lower probability lies with the principal of randomness and duplicates.

Since every combination is randomly generated independently from another, the chance that a newly printed number set is a duplicate increases as the number of combinations are sold. Eventually, it becomes more likely that new combination is a duplicate, rather than being a unique combination.

According to the LottoReport.com website, approximately 176 million tickets were sold in the Wednesday January 6th drawing and there was no jackpot winner. This is not surprising because there was only a 45.3% chance that a winning ticket would be sold.

Likewise, when 200 million combinations are sold, there is only a 49.6% chance of winning; and when 300 million combinations are sold, there is a 64.2 chance of wining  (but more than 1 in 3 chance of losing).

You can access our:

page to analyse the chances of winning the Powerball and other lotteries based on the number of tickets sold.

Saturday, September 5, 2015

When to Play Lotto Max or Lotto 649 Guide - Includes 2015 Lotto Max Changes + 2013 Lotto 649 Changes

Lotto Max or Lotto 649?
(from http://www.chfi.com/contests/win-with-lotto-649-lotto-max/)
The purpose of this post is to provide Canadian Lotto Max and Lotto 649 players with sufficient information to help them decide on which game they should spend their money.  As background, we have firs summarized the most recent changes to both of these games. This is followed by our player recommendation advice.

Lotto Max Modifies the Game
In July 2015, the first format change to Lotto Max was implemented. As changes go, this was very modest because the ticket price, number of balls, and number of prizes remained the same. Only the prize amounts of the top three second tier prizes (5/7, 6/7, and 6/7 + bonus) were lowered.

This was done in order to allow the Lotto Max Jackpot cap to be increased from $50 to $60 million.

No change was made to the MaxMillions feature, which begin to be offered at $50 Million. However, the number of MaxMillion prizes will be at the discretion of the organizers when the Jackpot level is between $50 and $60 million. In this range, they may increase the Jackpot, the number or MaxMillions, or both. Once the jackpot reaches $60 million, it remains fixed and only the number Max Million prizes increase.

Effect of 2015 Lotto Max Changes
From an executive viewpoint, three of the game characteristics remain unchanged.
  • The percent of winners remains at 5.05% 
  • The overall odds stays at 19.8
  • The return sensitivity stays constant at $0.006985.
But, the overall return is decreased to $0.353 (from $0.364), and payout per dollar spent is reduced to $7.00 versus $7.21.

Review of the Lotto 649 Changes in 2013
Since we first published our "When to Play Guide" for Lotto Max and Lotto 649 in 2009, we have not updated it since (although 649 was changed).

In September 2013, four significant changes were made to the Lotto 649 format:
  • First, the ticket price was increased to $3.00 (up from $2.00)
  • Second, the minimum jackpot prize was increased to $5 Million (up from $2.5)
  • Third, a new "Free Ticket" prize was added for those matching only 2 of the 6 main balls (not including the bonus ball).
  • Fourth, a separate $1 Million Guaranteed Prize drawing was added.
While the underlying concept of drawing 6 of 49 balls and one bonus ball remained the same, these changes produced very noticeable results, mainly:
  • The number of winners increased to 15.1% (up from 3.1%),
  • The overall odds dropped to 1 in 6.62 (from 1 in 32.31)
  • The overall return increased slightly to $0.333 per dollar (from $0.318)
  • The return sensitivity flattened to $0.023837 (from $0.035736)
  • And, the payout per dollar spent was cut to an average of $2.20 (from $10.27)

Break-even Jackpot
Using the Overall Returns and Return Sensitivities of the latest 2015 Lotto Max and the 2015 Lotto 649 games, we have calculated the break-even jackpot to be $4.15381 Million.

As shown above, the break-even amount is below both the minimum $5.0 Million Lotto 649 jackpot and the $10.0 Million Lotto Max minimum jackpot. Thus, for any level where the jackpots of both Lotto 649 and Lotto Max are equal, one should always purchase the Lotto 649 game because the return is higher.

However, we rarely have the situation when the jackpot offerings of both games is the same. This is because the Lotto Max minimum is $10.0 Million, and it increases by a minimum of $5.0 million, whereas the Lotto 649 minimum is only $5.0 and typically increases by just $2.0 to $5.0 million.

The "When to Play" Guide
Therefore, we need to view a cross-jackpot matrix to identify the various levels where it is more beneficial to play one game versus the other. Using the return information from the above graph, we have constructed a "When to Buy" graph showing the break-even Lotto 649 jackpots for Lotto Max jackpots ranging from $10.0 million to $100.0 million

Choosing the Game to Play - Our Advice
Based on the above, we see 10 vertical bars. The horizontal spacing indicates the Lotto Max jackpots labeled 10M to 100M. Immediately below them are the corresponding Lotto 649 break-even jackpots.

For example, we see that if the Lotto Max jackpot is $10 million, we should purchase this game whenever Lotto 649 is below $5.87 million. But, if the Lotto 649 jackpot is higher ($7M for instance), we should then buy Lotto 649.

Similarly, when Lotto Max is $50 Million, we should buy that game unless Lotto 649 is above $17.59 million.

Note that once Lotto Max reaches $50 Million, we must consider the total jackpot offering to determine the return., This is the total of both the advertised jackpot amount plus the total number of Max Millions. This is why we can consider Lotto Max jackpots from $50 to $100 Million.  For example, if the Lotto Max jackpot was $60 million plus 20 Max Millions, we would use the total of $80 million as the Lotto Max value when reading the graph above.

Lastly, to evaluate jackpot levels not displayed, one should simply use the values shown above and interpolate.

Friday, October 18, 2013

The October 19 2013 Mega Millions Lottery Changes

(from www.ctlottery.org)
Last Spring, Mega Millions announced that it was going to make major changes to its nationwide lottery game. This came in response to the Powerball price increase last year, and to the realization that the playing public is eager to frantically buy lottery tickets when the offered jackpots are absurdly high.

So to capitalize on the players desire to win gigantic jackpots and to derive larger revenue profits, the Mega Millions format was changed to make the probability of winning the jackpot even more difficult. By doing this, it is believed that the the average jackpot prize offerings will be in the $200 to $800 million range. And, by keeping the ticket cost at the same single dollar price, players will be more eager to purchase multiple tickets at once.

Truthfully, this logic is brilliant, and we believe that Mega Millions will accomplish its goals.

The 2013 Mega Millions Changes
The table below shows a summary of the 2013 Mega Millions changes. Most notably, players will find that the cost of ticket prices will remain the same $1 cost, prize payouts will be substantially lower, and the chances of winning will be much harder.

Summary of 2013 Mega Millions Changes Megaplier Prizes
  • White Balls increased to 1 to 75 (19 more)
  • Megaball lowered 1 to 15 (down from 46)
  • Starting jackpot at $15 million (up from $12 million)
  • Jackpot will increase by $5 million minimum
  • Match 5 prize fixed at $1 million (up from $250,000)
  • Regular Ticket Costs $1 (no change)
  • Chance of winning 258,890,850 (up 83M or 47.4%)
  • Annuity payout moved to 30 years (up from 26)
  • Annuity payments increase by 5% every year
  • Overall chance of winning any prize 14.7
  • Megaplier Ticket still Costs $2 (no change)
  • All prizes multiplied by 2, 3, 4 or 5 times (5 times is new)
  • Average megaplier value is 3.87 times

Because the number of white balls increased to 75, the chances of winning the 2nd tier prize of 5+0 has grown to 1 in 18,492,204. Comparing this to the old 1 in 3,904,701 value for 1 in 56, we see that it is 5.975 times harder to win this prize. This means that the the 2nd tier 5+0 prize is undervalued in the new Mega Millions game.

However, since the number of Megaballs decreases to 15, the overall chance of winning any prize drops to in in 14.7 verses one in 40 in the old format.

Start Date
The new Mega Millions will become effective on Saturday October 19, 2013, and the first drawing will be held on Tuesday October 22, 2013.

Mega Millions's Reasons for Changes.
According to summaries, the main reason that the Mega Millions format is changing is that they want to make the game more exciting by offering larger jackpots. To do this, they increased the number of combinations to make the jackpot more difficult to win. However, they wanted to be competitive with Powerball. So, they kept ticket price at the same $1 cost. Thus:
  1. to increase jackpots, they increased the odds of winning to 1 in 258.9 million;
  2. by keeping the price at $1, a player who spends $2 on Powerball can get 2 Mega Million Tickets;
  3. for the same Powerball price, the odds of winning are only 1 in 129.45 million (lower than Powerball);
  4. by offering the match 5 prize of $1 million, they are consistant with Powerball;
  5. by offering the Megaplier, a player matching 5 white balls only could win $2 to $5 million;
  6. by changing the annuity payment to 30 years, they are consistant with all the other major lottery annuities.
  7. and lastly, by lowering the Megaballs to only 15, the will be more overall winners.

Game Variety
Now that the odds of winning have increased by 47%, it is easy to see that winning Mega Millions will be more difficult. Further, the value of individual prizes has been reduced substantially.

To help understand the impact of the Mega Millions changes, the table below lists the Mega Millions and Powerballs differences side by side. Powerball is shown in the light blue columns, and the new Mega Millions is shown in the pink columns.

Match MM 5/75+1/15
PB 5/59+1/35
Prize $
Prize $
Prize $
Prize $
Power Play
5+1 258,890,850.00 175,223,510.00 1 1 $15,000,000 $40,000,000 $15,000,000 $40,000,000
5+0 18,492,203.57 5,153,632.65 14 34 $1,000,000 $1,000,000 $3,866,667 $2,000,000
4+1 739,688.14 648,975.96 350 270 $5,000 $10,000 $19,333 $40,000
4+0 52,834.87 19,087.53 4,900 9,180 $500 $100 $1,933 $200
3+1 10,720.12 12,244.83 24,150 14,310 $50 $100 $193 $200
3+0 765.72 360.14 338,100 486,540 $5 $7 $19 $14
2+1 472.95 706.43 547,400 248,040 $5 $7 $19 $14
2+0 33.78 20.78

1+1 56.47 110.81 4,584,475 1,581,255 $2 $4 $8 $12
1+0 4.03 3.26

0+1 21.39 55.41 12,103,014 3,162,510 $1 $4 $4 $12
0+0 1.53 1.63

Illustrated above are the odds (or chances), number of winners, prize payout for normal tickets, and prize payout for the multiplier (Megaplier and Power Play) tickets of both games.

Because the differences are rather large, we have decided not to illustrate these in graphic form.

In summary, we believe that the increased odds of winning both the Mega Millions Match 5 and Jackpot will result in a far fewer number of players winning prizes of  $1 million or more. But, since few people win these now, most players will not realize the importance of this improbability.

Correspondingly, players will obtain a "false" sense of security thinking that they really have a chance to win the jackpot because it seems there are more winners. Compared to Powerball, there will be 3 times more players who match the Megaball than those matching the Powerball number. But, those who win at Mega Millions will only receive $1 verses those Powerball players who receive $4.

So just think, you will get 1/4th the amount of money but 3 times the number of winners. So, in reality, things are much worse on a dollar to dollar basis.

But, those who play the Megaplier option will fare much better. In this case, one should compare the Megaplier payouts to the normal $2 Powerball ticket. When this is done, it is easier to see that Mega Millions odds are lower than Powerball and the payouts are much better.

However, we believe the new Mega Millions game will be very successful. Indeed, jackpots will be typically be large and the frequency of winning will keep players hooked on the game.

More Information:
The following links provide you with more information about the 2013 Mega Millions lottery.

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