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The Powerball jackpot for the Wednesday night draw could reach $ 448 million.
It's a very big price. However, looking more closely at the math behind the lottery, it is probably a bad idea to buy a ticket.
Consider the expected value
When trying to evaluate the outcome of a risky and probabilistic event such as the lottery, one of the first things to look at 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, then, in the long run, the game will earn me money. 39; money. If the expected value is negative, this game is for me a net loser.
The expected values combine probabilities and prices to give us a better idea of the value of a game. For example, consider a simple game in which you throw a coin. If the coin is a head, you pay me $ 2. If that's the tail, I'll pay you $ 1.
Intuitively, this game seems to be a bad idea for you, and the expected value shows why.
To calculate the expected value, we multiply the probability of each result by the value of each result and add them together. In our draw game, there is a one in two chance that you lose $ 2 and a 1/2 chance of winning a dollar, so your expected value is (1/2 × – $ 2) + (1 / 2 × 1 $), which gives us – $ 1 + $ 0.50, for a final expected value of – $ 0.50 for you.
Read more: Fractals are the most pretentious concept of maths, and they get even stranger when they are used to solve a puzzle involving the British coast.
Lotteries are an excellent example of this type of probabilistic process. In Powerball, for every $ 2 you buy, you pick five numbers from 1 to 69 in a white ball collection and a powerball red number from 1 to 26. Prizes are based on the number of numbers chosen by the player who 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 Powerball website usefully provides a list of odds and prices for possible game outcomes. We can use these probabilities and price sizes to gauge 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 -0.15 $. As it is less than zero, it would seem that the purchase of a Powerball ticket is not a good investment. But, looking at other aspects of the lottery, things get even worse.
Annuity versus lump sum
Just looking at the total price is a vast oversimplification.
First, the $ 448 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, but multi-million dollar, annual payments. 30 years.
If you choose instead to take the full price at once, you get a lot less money in advance: the cash payment value at the time of writing is 271.7 million of dollars.
If we take the lump sum, we end up seeing that the expected value of a ticket drops up to -0.75 $:
The question of whether to take the rent or the money is somewhat nuanced. The Powerball website says that annuity option payments increase by 5% each year, probably depending on inflation or beyond.
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 generally better to have more money today than to accept it over a long period, since a larger investment is now accumulating compound interest faster than smaller investments. 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 combined taxes of states, federal and, in some countries, local governments take up half the money.
Considering this fact, if we only win half of our potential earnings, our expected value calculations turn out to be more and more negative, suggesting that our investment in Powerball would be a bad idea.
Here's what we get by taking the annuity, after taking into account our approximate 50% tax reserve. The expected value drops to – $ 0.91:
Losing as much as half of the price due to taxes is just as devastating for the lump sum, bringing the expected value down to a whopping $ 1.22:
Even if you win, you could share the prize.
Another problem is the possibility of winning several jackpots.
Big pots, especially those that attract significant media coverage, tend to attract more customers with lottery tickets. And more people buying tickets mean more chances that two or more choose the magic numbers, which results 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 have seen the effect of reducing the jackpot by half taking into account the effect of taxes. Given the possibility of having to redo this, buying a ticket is almost certainly a losing solution 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 a lottery ticket, the number of repetitions is the number of tickets sold and the probability of success is 1 in 292,338 chance of obtaining 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 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 chances of a divided jackpot.
According to our analysis of LottoReport.com's recordings, Powerball's average draws represent no more than 18.3 million tickets. On average, up to now, the draw has totaled 18.3 million tickets, leaving only a chance of about 0.2% of a split pot. But higher prices could attract more players – the record $ 1.6 billion jackpot in October 2018 had sold about 370 million tickets before its last draw.
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 chances of A divided jackpot and damage the value. of a ticket.
Good luck to all who still play the lottery despite all this!
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