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The Mega Millions jackpot rises to $ 868 million at 9:45 am Wednesday, after no one won the Tuesday night draw.
(The Powerball jackpot also represents a respectable, although much lower, $ 345 million gain before the anticipated draw on Wednesday night.)
This is the biggest Mega Millions prize of all time, according to the lottery website. However, by looking more closely at the mathematical calculations underlying the lottery, it is probably wrong to buy a ticket.
Consider the expected value
When trying to assess the outcome of a risky and probabilistic event such as the lottery, one of the first things to consider is the expected value.
The expected value is useful for evaluating the results of the game. If my expected value for the game, based on the cost of the game and the odds of winning different prizes, is positive, the game will bring me long-term money. . If the expected value is negative, then this game is a net loser for me.
Lotteries are an excellent example of this type of probabilistic process. In Mega Millions, for every $ 2 ticket you buy, you choose five numbers from 1 to 70 and one from 1 to 25. Prizes are based on the number of numbers chosen by the player that match those drawn.
Match the six numbers and you win the jackpot. After that, there are smaller prices to match a subset of the numbers.
The Mega Millions website usefully provides a list of ratings and prices for possible game outcomes. We can use these probabilities and price sizes to evaluate the expected value of a $ 2 bill.
The expected value of a randomly determined process is determined by taking all possible outcomes of the process, multiplying each result by its probability and adding all those numbers. This gives us a long term average value for our random process.
Take each prize, subtract the price of our ticket, multiply the net return by the probability of winning and add all these values to get the expected value.
We are left with an expected value of $ 1.12, which is positive and above our break-even point. This suggests that it would be a good idea to buy a ticket – but taking into account other aspects of the lottery makes things go wrong.
Annuity versus lump sum
To look only at the total price is a vast oversimplification.
First, the $ 868 million jackpot is paid in the form of an annuity, which means that instead of receiving the full amount at one time, it is divided into smaller annual installments – albeit multi-million dollar – on 30 years.
If you choose instead to receive the full price at once, you get a lot less money in advance: the cash payment value at the time of writing is $ 494 million .
If we take the lump sum, we end up seeing that the expected value of a ticket falls below zero, at -0.12, suggesting that a ticket is a bad deal.
The question of whether to take the rent or the money is somewhat nuanced. The Mega Millions website indicates that annuity option payments increase by 5% every year, probably at the same rate as inflation.
On the other hand, the state invests liquidity somewhat conservatively, in a combination of government securities and US agencies. It is quite possible, though risky, to get a higher return on cash if invested wisely.
In addition, it is often better to have more money today than to accept it over a long period, since a larger investment today will accumulate compound interest faster than smaller investments made. over time. This is what is called the time value of money.
Taxes make matters worse
In addition to comparing the annuity to the lump sum, there is also the big tax reserve. Although state income taxes vary, it is possible that the combined taxes of states, federal and, in some countries, local governments take up half of the money.
If we take this into account, if we only win half of our potential earnings, our expected value calculations go further into negative territory, making our investment in Mega Millions an increasingly bad idea. .
Here's what we get from taking care of the annuity, after taking into account our back of the envelope estimated at 50% in taxes. Expected value drops to – $ 0.32.
The flat rate tax is just as damaging.
Even if you win, you could split the price.
Another problem is the possibility of winning several jackpots.
Big pots, especially those that attract significant media coverage, tend to attract more customers who buy lottery tickets. And more people buying tickets mean more chances that two or more choose the magic numbers, resulting in the equal distribution of the prize among all the winners.
It should be clear that this would be devastating for the expected value of a ticket. Calculating the expected values taking into account the possibility of multiple winners is tricky because it depends on the number of tickets sold, which we will not know before the draw.
However, we saw the effect of reducing the jackpot by half considering the effect of taxes. Given the possibility of having to redo this, buying a ticket is almost certainly a waste, if there is a good chance we need to share the pot.
One thing we can calculate quite easily is the probability of multiple winners depending on the number of tickets sold.
The number of winners of a lottery is a classic example of a binomial distribution, a formula of basic probability theory. If we repeat a probabilistic process a number of times and each repetition has a fixed probability of "success" as opposed to "failure", the binomial distribution indicates the probability that we have a certain number of successes.
In our case, the process involves filling out a lottery ticket, the number of repetitions is the number of tickets sold and the probability of success is 1 in 302,575,350 chance of getting a winning ticket.
Using the binomial distribution, we can find the probability of splitting the jackpot based on the number of tickets sold.
It should be noted that the binomial model for the number of winners has an additional assumption: lottery players choose their numbers at random. Of course, not all players will, and some numbers may be chosen more often than others. If one of these more popular numbers comes up for the next draw, the chances of splitting the jackpot will be slightly higher. Nevertheless, the chart above gives us at least a good idea of the odds of a split jackpot.
According to our analysis of LottoReport.com's recordings, the average Mega Millions draws do not pose many risks of multiple winners – the average draw in 2018 is up to approximately 19.2 million sold here. of tickets, leaving only a chance of about 0.2% to divide the pot. According to LottoReport.com, even Tuesday's draw, which yielded about 105.2 million tickets, left only a 4.8% chance of a shared pot, according to the distribution analysis. binomial.
The risk of splitting lots gives rise to difficulties: ever-increasing jackpots, which should give a better value expected for a ticket, could have the unexpected consequence of bringing in too many new players, increasing the odds of A divided jackpot and hurt the value of a ticket.
Good luck to all those who still play the lottery despite all this!
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