## Lotteries

A lottery is a type of gambling where people buy tickets, and then win if their numbers get chosen.

A “lot” is something that happens by chance. *You may have heard people say “let us decide by drawing lots” or “so that is my lot”.*

### Rules

Different Lotteries have different rules.

Here we will use a typical lottery where the player chooses **6 different numbers out of 49**.

#### Example:

You enter the lottery by buying a ticket and selecting your six numbers.

You choose: **1, 2, 12, 14, 20 and 21**

On Saturday they draw the lottery, and the **winning numbers** are:

3, 12, 18, 20, 32 and 43

You matched **two** of the numbers (12 and 20):

- Is that enough to win you anything?
**No.** - Usually you must match at least
**three**numbers to get a small prize. - Matching
**four**numbers gets a bigger prize, - Matching
**five**is even bigger. - But if you match
**ALL SIX**of the numbers you**might win millions**.

The chances of winning **all 6 numbers** is 1 in 13,983,816 (calculated below).

### Choosing Numbers

These might win.

The numbers don’t know what they are!

A Lottery is **just as likely** to come up “1,2,3,4,5,6” as “9,11,16,23,27,36”

Instead of numbers they could be symbols, or colors, the lottery would still work.

In fact the result below really happened (Florida Fantasy 5 on 21 March 2011):

So it doesn’t matter what numbers you choose, the chances are all the same.

### More Likely Numbers?

So you have read that some numbers come up more often than others? Well of course they do, that is random chance.

The people who run lotteries have strict rules to stop the “rigging” of results. But random chance can sometimes produce strange results.

For example, using The Spinner I did 1000 spins for 10 numbers and came up with this:

Wow! **7 came up 115 times**,

and 8 only 81 times.

Does this mean 7 will now come up *more often*, or *less often*? In fact it doesn’t mean anything, 7 is just as likely as any number to get chosen.

Try it yourself and see what results you get.

### Popular Numbers

**But there is a trick!** People have favorite numbers, so when popular numbers come up you are sharing the winnings with lots of people.

Birthdays are popular choices, so people choose 1-12 and 1-31 more often. Also lucky numbers.

So maybe you should choose **unpopular** numbers so when you DO win you get more money.

(This assumes your lottery is one where prizes are shared among winners.)

### Regret

**Don’t choose the same numbers every week**. It’s a trap! If you forget a week you then worry that your numbers *will* come up, and this forces you to buy a ticket every week (even when you have other more important things to do).

Make a list of many unpopular numbers.

Choose randomly from this list every time.

### Syndicates

A “Syndicate” is a group of people who all put in a little money so the group can buy lots of tickets. The chance of winning goes up, but your payout each time is less (because you are sharing).

Syndicates can be fun because they are sociable . a way of making and keeping friendships. Plus some syndicates like to spend small winnings on everyone going out for a meal together.

Another good reason for joining a syndicate is that your chances of winning go up (but what you win goes down).

Think about it . winning Ten Million would really change your life, but One Million would also improve your life a lot. You might prefer ten times the chance of winning one million.

### Chance Of Winning the Big Prize

OK. What are the chances of you winning the big prize?

The chances of winning **all 6 numbers** is 1 in 13,983,816

You can use the Combinations and Permutations Calculator to work it out (use **n=49**, **r=6**, ‘No’ for Is Order Important? and ‘No’ for Is Repetition allowed?)

The actual calculation is this:

49 C_{6} = 49!/(43! x 6!) = 13983816

So how many times do you need to play to win?

#### 1 Week

Suppose you play **every week**

The probability of a **win** after 1 week is:

*1* **13983816** = 0.0000000715.

So the probability of **no win** after 1 week is:

1 в€’ *1* **13983816** = 0.9999999285.

#### 50 Years

Let’s say you play for 50 years, that’s 2,600 weeks.

The probability of **no win** over 2,600 weeks is:

( 1 в€’ *1* **13983816** ) 2600 = 0.999814.

That means the **probability of winning** (after 50 Years) is: 1 в€’ 0.999814. = 0.000186.

Still only about **0.02%**

And you would have spent thousands for that small chance.

You may have had some good holidays for that money.

**BUT it IS fun thinking “I just may win this week!”**

Just keep it as a **fun** thing to do, OK?

### Your Turn

- Find out the rules for winning Lotto in your area.
- How many numbers do you have to choose and how many numbers do you choose from?
- Calculate the probability of winning in any one week.
- Calculate the probability of winning if you play every week for 50 years.
- How much money would you save by not playing? What can that money buy?

A look at lotteries from a mathematical viewpoint.

## 9 ways you can win money in the Powerball lottery

With a $1.4 billion Powerball lottery up for grabs, even those who don’t normally play the lottery are throwing their hats in the ring.

For the uninitiated Powerball players who may have impulsively purchased a ticket, we decided to break down all the possible ways you can win. You know, just in case.

First, the basics. The Powerball lotto is made up of 6 total numbers randomly selected twice per week. Every Wednesday and Saturday at 10:59 p.m. EST, a live drawing is held. All the numbers are printed on white or red balls, and placed into an automated machine which spits out the winning digits.

The first 5 numbers (white) are between 1 and 69. The final number (red) is called the Powerball. It can be anywhere between 1 and 26.

When purchasing a ticket, players can select all 6 numbers themselves or do a “quick pick” and receive a randomly generated set of numbers. Tickets are $2 each, but you can pay an extra $1 at the time of purchase to upgrade your ticket to “Power Play.”

The “Power Play” add-on means your cash prize will, at minimum, double if you have a winning ticket for anything other than the grand prize jackpot.

So, what exactly are the winning tickets (aside from the jackpot)? **The possible winning tickets and their cash prizes are as follows:**

- Match all 5 white numbers only — $1 million
- Match 4 out of 5 white numbers + Powerball — $50,000
- Match 4 white numbers only — $100
- Match 3 white numbers + Powerball — $100
- Match 3 white numbers only — $7
- Match 2 white numbers + Powerball — $7
- Match 1 white number + Powerball — $4
- Powerball only — $4

The “Power Play” bonus is announced at the drawing time, and does not apply to the jackpot winnings. If you’ve matched all 5 white numbers and selected “Power Play” at the time of your ticket purchase, you will win $2 million (instead of the standard $1 million). For all other prizes, the “Power Play” bonus with either be 2x, 3x, 4x, or 5x your winnings.

Overall odds of winning any of the prizes starting at $4 is 1 in 24.87. On the off-chance you do take home a lump sum, don’t forget to hire a lawyer.

The next Powerball drawing takes place Wednesday, January 13 at 10:59 p.m. EST.

**UPDATE:** The winning numbers drawn Wednesday were **8, 27, 34, 4, 19 — Powerball 10.** Reports indicate three jackpot winners will split the grand prize of $1.5 billion.

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The winning ticket prizes start at $4, so at least you'd get your money back.