La Lotería de Navidad / sPANISH CHRISTMAS lOTTERY

"The lottery is a tax on poor people and on people who can’t do math." David Ramsey.

Introduction

This is an activity related to probability, a very important branch of Mathematics.

To start, the teacher gives you three positives for you to play in the lottery below. You can win extra positives to play with if you answer the questions in grey correctly.

But first, let's watch a video.

The "Lotería de Navidad" is one of the most popular gambling games in Spain.

To start you need to buy a ticket ("billete") or a fraction of it ("décimo" or "participación"), associated with a number from 0 to 99 999.

Example of one tenth of a ticket - Ejemplo de décimo

1. What is special about that number?

There are 180 series of 100 000 tickets (numbers). Each tenth of a ticket costs 20 euros, but sometimes people share part of a tenth, which is called "participación".

Detalle de una participación

2. How much does the total of the tickets for sale cost?

3.6 billions €

The teacher offers five of those "shares" for you to choose one of them. The price of each "share" in our game is 1 positive. The prize in case you win is 10 positives.

3. If one person wants to buy one whole number, all the series. How much does he or she have to spend on it?

36000 €

1st "Sorteo de Navidad del IES Fray Luis"

Buy with one of your positives one of the "decimos" and play with it.

Did you get lucky? Don't worry, you'll have more chances.

The rAffle

But how is the real raffle?

100 000 small balls are introduced in a big rotating drum the lottery day. On the other hand, there is another, much smaller, rotating drum with 1807 prizes marked on other balls.

The kids from one school in Madrid, "Los niños de San Ildefonso", are in charge of pairing both balls, numbers and prizes, in a jolly and catchy jingle. Naturally, there are way more numbers than prizes. When a big prize comes up the show stops to let a public notary witnesses it.

Watch this video on the big prize:

The prizes

The first prize gives its name to the lottery: "El Gordo" (The Fat One). But there are many other prizes.

Find below the complete list of prizes. Those that are "sung" in the event appear highlighted. There are other prizes: for numbers close to "el gordo", the second or the third prize; reimbursements for numbers ending in the same digit than "el gordo"...

The prizes are given to every series, that is, a ticket (ten tenths).

4. How would the owner of the share (participación) above win if that number was the first prize?

5. What if he/she owns two tenths (décimos) of the number 88302?

6. What is the prize per euro if you get a reimbursement?

4. 4 000 000 € : 200 €/t = 20 000 each euro, then 20000 · 5 = 100 000 €

5. 20 000 · 40 = 800 000 €

6. 1 euro per euro.

2nd "Sorteo de Navidad del IES Fray Luis"

Now we have some other prizes to make the raffle more realistic.

1 first prize of 10 positives.
1 second prize of 5 positives.
1 third prize of 3 positives.
Reimbursement for all the numbers that end in the same digit as the first prize.

How does a mathematician analises the lottery?

This kind of experiments are called random experiments ("experimentos aleatorios"). In those experiments, there are more than one possible outcome. The probability of an event is a measurement of the likelihood that the event will occur.

In Mathematics those experiments are measured with percentages (or a number between 0 and 1). A probability or 100% means that the experiment will occur for sure; a probability of 0% means that it is impossible for it to happen.

You can find the probability of an event A using the following formula. Remember that a number like 0.75 is equal to 75%. So the probability of a random event is always a number between 0 and 1.

Example: if you spin a coin, there are two possible outcomes, heads or tails. If the coin is fair, or unbiased, eacho of these outcomes is equaly likely. Taking a head as de "successful" outcome, P(H) = 1/2.

7. What is the probability of getting at least one heads when tossing two coins?

8. Calculate the percentage of prizes according to the table above. What is the probability of a ticket to get a prize?

9. Calculate the probability of getting the big prize.

3rd "Sorteo de Navidad del IES Fray Luis"

Now we have some other prizes to make the raffle more real. Also, you can choose any number you want (make a big drawing of it on your notebook).

1 first prize of 10 positives.
1 second prize of 5 positives.
1 third prize of 3 positives.
Repayments for all the numbers that end in the same digit than the first prize.

For this third raffle you can choose between buying your "participación" or not.

Other lottery games and the odds to win

These are other lottery games and the odds to hit the jackpot [*]:

La Quiniela: guess the result of 14 soccer games (wins, losses and ties)
- Odds: 1 to 5 000 000 aprox.
- Wins: a list of 14 results like this example 1-1-X-2-1-X-1-X-2-1-X-1-1-1-1
- Possible: ¿? 3⁵
Lotería primitiva and bonoloto: You have to hit 6 numbers from 1 to 49.
- Odds: 1 to 13 983 816.
- Wins (example): 1, 2, 3, 4, 5, 6
- Possible: ¿? 49C6
El cuponazo: A five-digit number, plus the series (from 000 to 999).
- Odds:
- Wins (example): 11235 813
- Possible
Mega Millions (USA): pick 5 numbers from 1 to 70 ($2 per play) and 1 Mega Ball from 1 to 25. It is played in 45 USA states.
- Odds:
- Wins (example): 1, 2, 3, 5, 8 and 13.
- Possible: 70C5 · 25
Florida lottery: 1 to 22 957 480.
El gordo de La primitiva: 1 to 31 625 100. Hit 5 numbers from 1 to 54 and 1 more from 0 to 9.
Euromillones: 1 to 76 275 360. Hit 5 numbers from 1 to 50 and 2 "stars" in other box, from 1 to 11.

Some final questions

How many positives did you have at the beginning?
How many positives do you have now?
What would have been better for you to play or not to play?

Para debatir

Hay gente que pasa más de dos horas haciendo cola para comprar décimos de lotería en una administración de Madrid.

¿Qué opinas de eso?

¿Qué número es más bonito, el 45137 o el 00000? ¿Por qué?

¿Qué número tiene más posibilidades de ser premiado, el primero o el segundo?

Para ampliar

Una persona gasta 200 euros en lotería de navidad cada año. Lo hace desde que tenía 18 años y ahora tiene 70. ¿Cuánto dinero se ha gastado? ¿Cuánto tendría en la actualidad si hubiese invertido ese dinero cada año en un plazo fijo al 2%?