**The Mega Millions jackpot hit a record $ 868 million after no one won the drawing on Tuesday.****Although that's a pretty big deal, buying a ticket is not a good investment.****The low probability of winning and the risk of sharing the price in a large, high-stakes game means that you are likely to lose money.**

(The Powerball jackpot is also a respectable, albeit much lower, $ 345 million pre-draw draw on Wednesday night.)

This is the biggest Mega Millions award ever on the lottery's website. However, a closer look at the lottery's underlying mathematics shows that buying a ticket is probably a bad idea.

## Consider the expected value

If you are trying to evaluate the outcome of a risky, probabilistic event like the lottery One of the first things to look at is the expected value.

The expected value is useful for judging the results of gambling. If my expected value for playing, based on the cost of the game and the odds of winning different prizes, is positive, then the game will bring me money in the long run. If the expected value is negative, then this game is a net loser for me.

Lotteries are a great example of such a probabilistic process. In Mega Millions, you choose five numbers from 1 to 70 and one from 1 to 25 for each $ 2 lot you buy. Pricing is based on the number of players you have selected.

Find all six numbers and win the jackpot. After that, there are smaller prices for adjusting a subset of the numbers.

The Mega Millions website helpfully provides a list of odds and prizes for the possible outcomes of the game. We can use these probabilities and price sizes to evaluate the expected value of a $ 2 ticket.

The expected value of a randomly decided process is found by taking all possible outcomes of the process, multiplying each result by its probability and adding all these numbers. This gives us a long term average for our random process.

Take any price, subtract the price of our ticket, multiply the net yield by the probability of winning, and add all these values to get our expected value. 19659015] v2 1 input tax annuity "class =" JsEnabled_Op (0) JsEnabled_Bg (n) Trsdu (.42s) Bgr (nr) Bgz (cv) Maw (100%) "itemprop =" url "style =" background-image: url (https://s.yimg.com/ny/api/res/1.2/xPoXcvQ3xzWrXMboFTbtYQ–~A/YXBwaWQ9aGlnaGxhbmRlcjtzbT0xO3c9ODAw/http://media.zenfs.com/en-US/homerun/businessinsider.com/c6bab0deb0e45a9c35ab7e8976b47e9e); "src =" https://s.yimg.com/g/images/spaceball.gif "data-reactid =" 32 "/>

<p class =" Canvas Atom Canvas -Text Mb (1.0em) Mb (0) – sm Mt (0.8em) – sm "type =" text "content =" Business Insider / Andy Kiersz, Opportunities and Prizes of Mega Millions "data-reactid = "48"> Business Insider / Andy Kiersz, Mega Millions Opportunities and Prizes [19659018] In the end, we expect an expected value of $ 1.12, which is positive and exceeds our break-even point suggesting that it makes sense to buy a ticket – but considering other aspects of the L otterie does it wrong.

## Annuity versus lump sum

Considering only the grand prize, this is an enormous simplification.

The $ 868 million jackpot is paid as an annuity, which means that the total is not distributed all at once, but in smaller – but still multi-million – annual payments over 30 years.

If you choose instead, the Cash Payout Value at the time of writing is $ 494 million.

If we accept the lump sum, we see that the expected value of a ticket drops below zero, to – $ 0.12, suggesting that a ticket is a bad deal.

The question of whether to take the pension or the money is a bit nuanced. The Mega Millions website states that annuity option payments increase by 5% each year and are likely to keep pace with or exceed inflation.

On the other hand, the state is investing the money somewhat conservatively, in a mix of US government and agency securities. It is quite possible, albeit risky, to achieve a higher return on the cash amount, if it is meaningfully invested.

In addition, today it is often better to have more money than to take money over a long period of time, since today a larger investment is pooled interest faster than smaller investments made over time. This is called the time value of money.

## Taxes make things much worse

In addition to comparing the annuity with the lump sum, there is also the big restriction on taxes. While state income taxes vary, it is possible that combined state, federal and, in some jurisdictions, local taxes may claim up to half of the money.

Dividing this, if we only take half of our potential home prices, our expected value calculations move deeper into negative territory, making our investment in Mega Millions an increasingly bad idea.

Here's what we get from the annuity after taxing our estimated 50% envelope. The expected value drops to – $ 0.32.

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The flat-rate tax credit is just as damaging.

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## Even if you win, you can share the prize

Another problem is the ability to win several jackpots.

Larger pots, especially those that attract a lot of media attention, tend to attract more lottery customers. And more people who buy tickets have a greater chance that two or more will choose the magic numbers, causing the prize to be evenly divided among all winners.

It should be clear that this would be devastating for the expected value of a ticket. The calculation of expected values taking into account the possibility of multiple winners is difficult, as this depends on the number of cards sold, which we only know after the draw.

However, we have seen the effect of cracking the jackpot taking into account the impact of taxes. Given the opportunity to do so again, buying a ticket is likely to be a loss rate if there is a good chance of sharing the pot.

One thing we can relatively easily calculate is the probability of multiple winners over the number of tickets sold.

The number of jackpot winners in a lottery is a textbook example of a binomial distribution, a formula from the baseline probability theory. If we repeat a probabilistic process a few times and each iteration has a certain probability of "success" as opposed to "failure," the binomial distribution tells us how likely we are to have a certain number of successes.

In our case, the process fills in a lottery ticket, the number of iterations is the number of tickets sold, and the likelihood of success is the 1: 302.575.350 chance of winning a jackpot ticket.

Using the binomial distribution, we can find the likelihood of splitting the jackpot based on the number of tickets sold.

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It & nbsp; In view of the fact that the binomial model for the number of winners has an additional assumption: that the lottery players choose their numbers randomly. Of course, not every player will do so, and it's possible that some numbers will be chosen more often than others. If one of these more popular numbers appears on the next drawing, the odds of sharing the jackpot are slightly higher. Nevertheless, the above graph gives us at least a good idea of the chances of a split jackpot.

Most Mega Millions drawings are not risky for multiple winners – the average profit margin in 2018 so far has been around 19.2 million tickets. According to our analysis of LottoReport.com's records, there is only a 0.2% chance a split pot. Even Tuesday's draw, which, according to LottoReport.com, contained around 105.2 million tickets, only had a 4.8% chance of a split pot based on binomial analysis.

The risk of splitting prices leads to a mystery: Increasing jackpots, which should lead to a better expected value of a ticket, could have the unintended consequence of bringing in too many new players, increasing the chances of a shared jackpot and to damage the value of a ticket.

To all still lotto game in spite of everything, good luck!

NOW CLOCK: Never take the & # 39; clean & # 39; Part of moldy bread

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