At the moment, The Powerball jackpot has hit $600 million. This is a huge jump, and what it means is that a lot of people will start to play the lottery.
Before diving in to play, there are three statistical facts you need to know to figure out if it's how you want to spend your money.
Here are the probabilities for winning Powerball, according to their official site:
Let's see what we can figure out from this.
1. Expected Value
So the first thing we want to calculate is the expected value.
Let's say you have four dollar bills of different denominations in a bag. You can't see them, but I tell you that one is a twenty, two are singles, and one is a five. I'm charging you six bucks to reach in and grab a bank note. Should you play?
Well, the probability of drawing any of the four bills is 0.25. There are two singles, so the probability of drawing a single is 0.5. If we multiply the probability and payout of each event, and sum that for all events, we get the expected value of playing the game.
Here's the expected value for the money game. Keep in mind I'm charging you $6 to play:
EV = ($20—$6)*(0.25) + ($5—$6)*(0.25) + ($1—$6)*(0.5)
EV = ($14*0.25) + (-$1*0.25) + (-$5*0.5) = $0.75
So, on average, you'll make seventy five cents by playing the game. So you should, theoretically, play.
Let's check that out with the current Powerball game, plugging in the jackpot and using the probabilities given.
Walter Hickey / BI
So the expected value of playing the game is winning $1.78. That's not bad, but it's pretty important to know that almost all of that is derived from the ludicrously inflated and highly improbable jackpot.
Keep in mind, this doesn't account for the possibility of multiple winners splitting the jackpot, which should be a crucial part of any serious return on investment analysis.
2. Odds Of Any Winner
Last time the Powerball was this high, we figured out the probability of a player winning the game on any number of draws.
Given the astronomical 1 in 175 million odds, it's pretty unlikely that you'll be the winner. But for there to be one — or multiple — winners, it's not that uncommon.
The probability that there is not a single Jackpot lottery winner in a drawing is defined by this formula:
Probability of no winner = (1 - Jackpot probability)Number of Tickets
Which means that the probability of one or more winners is:
Probability of winner(s) = 1 - [(1 - Jackpot probability) Number of Tickets]
Since we don't know the eventual number of tickets, here's a chart that shows the different probabilities for different values:
Walter Hickey / BI
There are even odds of a winner at the point when around 120 million tickets are sold.
3. How many tickets should I buy to guarantee winning a prize?
So this doesn't prove you'll make a profit, but if you crave the sensation of success and want to guarantee it, here's how many tickets you should buy to guarantee — with certain degrees of confidence — that you'll have at least one winning ticket for some prize.
This is similar to the last problem, only the calculation looks like this:
P(At least 1 winning ticket) = 1 - [(Probability of not winning a prize) Number of Tickets ]
From the Expected Value problem, recall there is a 96.9% chance you have a losing ticket.
Here are the probabilities of at least one winner for any number of bought tickets:
Walter Hickey / BI
So, some notes:
- When you buy 9 tickets, you're estimated to win at least one prize 25% of the time
- At 13 tickets, you're estimated to win at least one prize 33% of the time
- At 22 tickets, you're estimated to win at least one prize 50% of the time
- At 44 tickets, you're estimated to win at least one prize 75% of the time
- At 73 tickets, you're estimated to win at least one prize 90% of the time
- At 95 tickets, you're estimated to win at least one prize 95% of the time
Just to give you a perspective on how hard it can be to be certain, you'll need to buy 145 tickets to be 99% sure that you have at least one winning entry.
Keep in mind, this analysis is if you assumed the quickdraw random selection. If you want to guarantee one win, you could just buy 35 tickets, each with a unique powerball. That would guarantee you win $4 minimum.
However, we want to know how it looks with a random selection.
May the odds be occasionally in your favor.
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