Thursday, 27 August 2015

Gambling and Expected Value: NHL Lotto (OLG)

In this post on Gambling and Expected Value, we look at the "NHL Lotto" lottery offered by the OLG.
Click here to find similar posts on other lotteries and games of chance.

NHL Lotto (OLG)

NHL Lotto a lottery offered by the Ontario Lottery and Gaming Corporation (OLG). 

How the Game Works

To play NHL Lotto, the player selects a combination of 5 numbers chosen from 1 to 30, and an NHL team (of which there are also 30 options). Numbers cannot be repeated and the order is not important. The player also has the option of having the computer generate a random selection instead (known as "Quick Pick" in lottery vernacular). The game involves two portions, the "Instant Win" and the "Nightly Draw". In the Instant Win portion, you watch a computer-generated animation of a hockey player take three shots at a net with targets and hope that you've been randomly selected to be a winner. The system has been programmed to randomly mete out prizes based on OLGs predetermined odds of winning. In the Nightly Draw, prizes are awarded based on correctly matching the winning numbers and NHL team. It costs $2 to play NHL Lotto. 

Probabilities and Prizes

There are 13 different ways to win a prize in the Instant Win portion of NHL Lotto, ranging from $15,000 down to $2. The prize awarded corresponds to the targets hit by your computer generated hockey player, but it's not clear how exactly that translates into calculable probabilities. I'll have to assume the odds reported by the OLG are the exact probabilities programmed into their computers. But here's the summary of the Instant Win prizes and probabilities for NHL Lotto:

Summary of prizes and their probabilities in the Instant Win portion of NHL Lotto

There are 142,506 unique combinations of five numbers that can be chosen from the numbers 1 to 30. The number of combinations of items you can choose from a set without repetition is calculated using the following formula:

 \binom nk = \frac{n!}{k!(n-k)!},

where k is the number of items chosen, n is the total number of item to choose from, and ! denotes the factorial function. In this case, n = 30 and k = 5. But you also choose one hockey team out of 30 possible choices. You could repeat the above calculation for n = 30 and k = 1, but it should be intuitively obvious that there are 30 possible ways to pick one hockey team out of the 30. So in total there are 30 times 142,506 = 4,275,180 unique combinations of one NHL team plus 5 numbers chosen from the numbers 1 to 30.

The probability of matching some number of the winning 5 can be calculated from the hypergeometric distribution. The formula looks like this:

Equation 1 (general lottery prize probability formula)

where p is the probability of winning, n is the size of the set of numbers the winners are drawn from, k is the size of a combination, and r is the number of matching numbers required to win the prize. For example, to calculate the probability of matching four numbers in NHL Lotto, r = 4, k = 5, and n = 30. Plugging in the numbers gives the probability of approximately 0.088%, or about 1 in 1,140 (note this probability reflects the combined probability of winning either the 3rd or 4th prizes).

We can treat the NHL team matching as a bonus number drawn from a separate pool of numbers. The formula to calculate the probability of matching r out of k winning numbers plus a bonus number that is drawn from a separate pool of m numbers is a slight modification of Equation 1:

Equation 2 (general lottery with separate bonus number prize probability formula) 

Here is a table summarizing the payouts of the Nightly Draw portion of NHL Lotto and their probabilities:

Summary of prizes and their probabilities for the Nightly Draw portion of NHL Lotto.

Since for some of the prizes matching x out of 5 numbers has two different prizes depending on whether or not you match the NHL team, the probability has to be adjusted for overcounting. The probability of matching with the bonus is calculated from Equation 2 (m = 30). The probability of matching without the bonus is calculated from Equation 1 minus Equation 2.

So there's roughly 66.7% chance you lose on the Instant Win game and 94.5% you lose on the Nightly Draw. The combined probability of losing both games is calculated by multiplying the two probabilities together, which works out to a 63.1% probability that an NHL Lotto player has just thrown his money away.

Expected Value

Expected value here is pretty straightforward to calculate, just multiply each of the possible prizes by its respective probability, sum all the products, and then subtract the cost of the ticket. For example, the $100,000 prize in the Nightly Draw multiplied by the 1/4,275,180 probability of winning contributes an expected value of $100,000/4,275,180 (about 2.34 cents) to the overall expected value of an NHL Lotto ticket.


Summary of expected values of the prizes in the Instant Win portion of NHL Lotto

Summary of expected values of the prizes in the Nightly Draw portion of NHL Lotto
with overall expected value and house edge of NHL Lotto

From the tables, you can see there's a house edge of about 41.3%, meaning an average of 41.3 cents from every dollar spent on NHL Lotto goes straight to the OLG.

It's interesting to compare the two games. The total expected value of the Instant Win prizes is more than 4 times the expected value of Nightly Draw prizes. Similarly, the probability of getting a payout in the Instant Win portion is about 6 times the probability of getting a payout in the Nightly Draw. The OLG cleverly use a little computer animation, which is probably slightly entertaining to watch, and the fact that you get a payout about a third of the time, to hook people on the game. Most people appreciate that the chance of winning exists only if they buy a ticket, but don't really appreciate how minuscule that chance might be. The Nightly Draw is there to give players hope. The thinking is that if you lose the Instant Win game, well at least you've got another shot at free money. Too bad your chance of winning anything in the Nightly Draw is much lower than in the Instant Win. The Nightly Draw is even assigned a bigger top prize, because when the payout gets bigger, people tend to be more willing to make bad decisions (like buying another ticket).

From our analysis we can draw the following conclusions:

  1. NHL Lotto has a house edge of about 41.3%, which is a little better than a 50/50 draw. In other words, on average, OLG keeps 41.3 cents of every dollar spent playing NHL Lotto. Better than a lot of lotteries, but worse than casino games.
  2. The Instant Win portion of the game is significantly better than the Nightly Draw portion, both in terms of the expected prize value (about 4 times that of the Nightly Draw), and the odds of winning a payout (about 6 times higher chance of winning than in the Nightly Draw).
  3. The combination of flashy animation, small and relatively frequent payouts, and a second chance game that has a bigger top prize (but rarely pays out to anyone) is purposefully crafted to attract and hook gamblers on NHL Lotto. 

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  64. My name is Robert Evans, I am very grateful sharing this great testimonies with you, The best thing that has ever happened in my life is how i won the lottery. I am a man who believe that one day i will win the lottery finally my dreams came through when i email Dr.Noruwa and told him i need the lottery numbers. i have come a long way spending money on ticket just to make sure i win. But i never knew that winning was so easy until the day i meant the spell caster online which so many people has talked about that he is very great in casting lottery spell, so i decided to give him a try. I contacted this man and he did a spell and he gave me the winning lottery numbers. But believe me when the draws were out i was among winners. i won 70,912,673.20 million Dollar. Dr.Noruwa truly you are the best, with these man you can win millions of money through lottery. i am so very happy to meet these man, i will forever be grateful to you. Email him for your own winning lottery numbers his Email: doctornoruwavickyhealingtemple@gmail.com Or chat him on Telegram/ WhatsApp: +1(562)774-3626

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  67. BEST LOTTERY SPELL BY POWERFUL DR. OBINNA

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  77. Hi everyone, after 10 years of playing lotto game i am finally living my dream life, all thanks to Dr Dominion i will never forget that great man never in my entire life, i am who i am today all because of him he gave me a winning numbers to play and he prepared a lotto winning spell for me that helped me won my mega million. thank you so much Dr Dominion for what you have done for my entire generations, i will forever be grateful.
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  81. GET YOUR EX LOVER BACK NOW OR GET YOUR BROKEN RELATIONSHIP, MARRIAGE RESTORED.

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  83. TO WHOM IT MAY CONCERN DR LOJA IS REAL.

    my name is Bemjamin from UK (London) my wife has heard rumors about me and this issues made her filed for divorce and everything went up up and this case and the thought of losing my family now has kept my heart unhappy so i prayed for a solution and a way i can be able to bring back my wife and kids back and prove to her all these things she heard about me and the other lady working in my office are all lies because deep down in my heart i knew i never and will never cheat on my wife. to cut this whole story short my wife sister from the USA heard all the issues in ground and immediately asked me to contact DR LOJA she gave me his contacts details and i did as she said and DR LOJA told me everything will be fine in two days in two days my wife will come home for good again and this was exactly what happened and for this reason i say this man is real and can also help anyone with any of the below problem

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  84. Good day everybody, This is my testimony on how i won $4,200,201 million I want to use this opportunity to thank Great Priest Salami for helping me to win the lottery of $4,200,201 million dollars on Mega millions lottery ticket. I have been playing the lottery for the past 10 years now and I have never won. Ever since then I have not been able to win and I was so upset and I needed help to win the lottery. So I decided to go online and search for help, there i saw so many good testimonies about this man called Great Priest Salami of how he has cast lucky spell lotto for people to win the lottery. I contacted him also and told him I want to win a lottery, he cast a spell for me which I use and I played and won $4,200,201 million dollars. I am so grateful to this man, just in-case you also need him to help you win, you can contact him through his email: purenaturalhealer@gmail.com WhatsApp number +2348143757229

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  85. I'm Thomas Yi from the United States. I started playing lottery games 3 years ago and I have never won big. I went online to seek help on how I can win big in my lottery games and I saw some nice reviews about Dr Amber who has made different people huge winners in their lottery games with his prayers. I gave it a try and I contacted Dr Amber who told me how and what to do before I can become a big lotto winner and I accepted. He made special prayers for me at his temple and gave me the required numbers to play the lotto game and when I played it, I won a massive $235.4 million POWERBALL jackpot. I was so happy and I chose to receive my winnings in a one-time, lump-sum payment of $160,038,447.27 at Florida Lottery Headquarters in Tallahassee and the balance was given to the store that I bought & played the game from. I would love other people to seek help from him and he can be contacted through WhatsApp/call/text via this number +1 (406) 646-2222 or you can email him on: amberlottotemple@yahoo.com

    ReplyDelete
  86. I'm Melvin Martins. from the United States. I started playing lottery games 7 years ago and I have never won big. I

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    told me how and what to do before I can become a big lotto winner and I accepted. He made special prayers for me at his

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    POWERBALL jackpot. I was so happy and I chose to receive my winnings in a one-time, lump-sum payment of $160,038,447.27

    at Florida Lottery Headquarters in Tallahassee and the balance was given to the store that I bought & played the game

    from. I would love other people to seek help from him and he can be contacted through WhatsApp/call/text via this

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  87. There is a great PSYCHIC that gives lottery winning numbers to people. I won $739,342,000 Million dollars with the help of his winning numbers, I have been playing lottery games for years and I have never been lucky but my Friend in the office won 496,800.000 million dollars and resigned from work so I beg him for help and he said lord Bubuza predicted his winning after casting a spell and gave him the winning number to play and he won so I spoke to lord Bubuza on WhatsApp: +1 984 240 7211 for help. He responded and said a reading had to be done and a spell cast to reveal my SIX winning numbers, I provided his requirements for the reading and casting of the spell. After casting the spell, he gave me SIX numbers and said it’s my winning number and told me when and where to play it. I did as instructed and 24 hours later, I got a call that I won $739,342,000 million dollars, I thought it was a joke until the amount hit my bank account. Lord Bubuza can help you win too. Contact him via WhatsApp: +1 984 240 7211 or email: lordbubuzamiraclework@hotmail.com

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  88. Lottery is something majority of the people in the world want to win, not only winning the lottery, but they want to win big, i have been playing the lottery for couple of years now, but i haven,t really win big, personally i always think there is something behind all those winners who always win big, because my intense always tell me that every thing is possible, i manage to get in contact with one of the biggest lottery winner in the US, i asked him about the success, and he referred me to a very powerful voodoo doctor, (Dr Emmanuel), he is indeed a great man who i explained to , without no hidden details , (i really want to win the lottery so bad ) he put to work and few days latter he gave me numbers that never fails , these numbers was what i played and i won $1,000,000,00, 1 Million Dollars , (Dr Emmanuel) is blessed , you too can change your life too with his help, contact Email: dremmanuellottoryspellcaster@gmail.com

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  89. Are you having issues in your relationship, or finding it difficult to have a child of your own or are you the type having any diseases in your body. I will advice you to contact Dr. Abolo of Abolospell@gmail.com . The potency of his spell is so effective and can reunite broken relationship and can also make one get pregnant know matter the situation and not only that, he also has the power to cure all type of diseases within the period of 48hours.. Contact him today and thank me later.



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    You can also send him a MSG on whatsapp directly via: +447535135409.

    ReplyDelete
  90. Worry no more about how to win big in your lottery game for Dr Amber is able to give you sure winning numbers to win big on your lottery games for I am a winner of $360,000,000. Email: amberlottotemple@yahoo.com or WhatsApp him on +1 (318) 306-5044 for more information.

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  92. I never thought I would be known for winning the lottery in NEW YORK. I am so grateful to DR AMBER for all he has done for me. I wouldn't have been able to win a CENT on the lottery if not for the help of DR AMBER who gave me perfect winning numbers to play the lottery that made me the winner of $10,000,000 DOLLARS in 2019. I got in touch with him again on his webpage: [ https://amberlottotemple.com ] in request for sure winning numbers to play the lottery. He told me what to do and I did all he instructed & required to get the spell prepared for me and within 3 days, he gave me the perfect numbers to play the lottery and could you believe that I won $10,000,000 DOLLARS after playing the lottery making it the second time I have won and I haven't finished spending the last one I won in 2019. DR AMBER is truly a God sent and I highly recommend him to anyone that needs help to win the lottery. You can message him on WhatsApp with +1 318-306-5044 or email: amberlottotemple@yahoo.com

    JUAN HERNANDEZ FROM UNIONDALE NEW YORK, USA.

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  95. Good day to you reading this now. I am very happy to share my insight working with DR AMBER because his powerful spell made me financially buoyant that no bill scares me anymore after winning 87 MILLION DOLLARS from the lottery game I played with your sure winning numbers. I want to thank DR AMBER for his endless help and he's from the United States. For urgent response, Call or text +1 808 481 5132 or email: amberlottotemple@yahoo.com or visit: AMBERLOTTOTEMPLE.COM

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  96. Hello folks, I want to share my story of how I met Dr Amber who helped me win the lottery. Before I met him, life was hard for me. I love playing the lottery but I lost all my money to it. I became suicidal because I owe the bank lots of money. So, I went on the internet about how I can win the lottery and I got good advocacy about Dr Amber and his work. I got in touch with him and I told him what I’ve been through and he assured me that he will help me. I was a bit scared but I said ok. Dr Amber made a spell for me that cost me some money to get it done and I was given some accurate digits by him to play the lottery on my next game which I did as instructed by him. I was astonished when I got announced as the winner of $109 million dollars on the Mega Millions Lottery I played. I am grateful to Dr Amber for Indeed, he deserves the hype he gets from people. You can email him or visit his webpage on how to get yours done. Email: amberlottotemple@yahoo.com or visit: amberlottotemple.com

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  101. I never thought I could be this wealthy, after all I've been through trying to raise money and take good care of my family. I used to play the national lottery but have never been lucky! One day I read an article by someone named "Bran's mahan", on the internet how he was guided by a Man and became a lottery winner, I decided to contact the spellcaster to help me win too. He gives me full assurance that he will help me win the lottery, but I've to meet his requirements which I did. After some days he gives me some numbers to play. Which I played! I was in my sitting room when I read the notification and it said I had won £1M but I didn't really believe it. I quickly picked up my phone and called the national lottery and even when I heard the word ' Congratulations you're a winner' - I just couldn't take it in. My wife came down from upstairs when she saw me in tears on the phone. She thought I'd been scammed or something bad had happened though it was tears of joy, that very day was a blessing to me. On my way to tell my family members about my win, I discovered I had won £5 on another national lottery ticket I bought on a whim, all thanks to God and Dr Marcus for making my dreams come through. Reach Dr Marcus on his Whatsapp+2348110492028 on email drmacusspellcaster@gmail.com and I believe he will guide you too.

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  104. You can't win the lottery game without the help of voodoo and this is a fact most people has failed to accept, unfortunately i was one of them until i gave a try trust me if not for security reasons i would have made it public but nevertheless it is not too late to get in contact with Priest Aziba who gave me the spacial lucky numbers i used in playing just once and i won hug amount of money, i would advise you give a try, its not harmful neither is it a sin, its what Africans believe inn and it has also work for me here in the state. VOODOO SPELL LOTTERY NUMBER. Get in contact with Priest Aziba Via Email Priestazibasolutioncenter@gmail.com or WhatsApp him on +2348100368288 so you too can become a millionaire.

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