# Odds and Prizes for Mega-Sena Lottery

TR Smith is a product designer and former teacher who uses math in her work every day.

## How to Calculate Mega-Sena Probabilities

Mega-Sena is the national lottery of Brazil and is operated by the Caixa Econômica Federal Bank. It's the largest lottery in Brazil and tempts players with huge prizes. The structure of the game is different from that of lotteries in the US, Canada, and Europe. It has some similarities to keno. During the official lottery drawing, 6 numbers are selected from between 1 and 60. If you match all 6 you win your share of the grand prize, and if you match 5/6 or 4/6 you can win a smaller prize.

What makes Mega-Sena different from lotteries such as Powerball, Mega Millions, Canada's 6/49, EuroMillions, UK Lotto, Oz Lotto, and other games is that players can select 6 to 15 different numbers per board. The more numbers you select the better your odds of winning, but also the higher the cost of the board. The cost is proportional to how many groups of 6 there are. For example, if you pick 9 numbers, there are 84 different groups of 6, so the cost of the board is 84 times the base price. Your probability of winning depends on how many numbers you choose to play, which makes the calculations a little more difficult. See table below. In this respect, Mega-Sena is similar to the Philippines UltraLotto.

A common question asked by Mega-Sena lottery players is whether it is better to buy a single board with multiple numbers, or to spend the same amount of money on separate boards with fewer numbers. The answer is the probabilities.

## Cost of Buying More Than 6 Numbers

Numbers per Board
Number of Groups of 6
Price Increase Factor
6
1
1
7
7
7
8
28
28
9
84
84
10
210
210
11
462
462
12
924
924
13
1716
1716
14
3003
3003
15
5005
5005

The table above shows the price increase factor for every extra number you buy. The price increase factor is equal to the number of groups of 6. If you choose K numbers, there are

K(K-1)(K-2)(K-3)(K-4)(K-5)/720

different groups of 6. For example if the base price of a Mega-Sena lottery ticket with 6 numbers is \$2, then a board with 15 numbers costs 5,005 times that amount, or \$10,010. If the base price is \$3 per board, then a board with 9 numbers costs \$3 × 84 = \$252.

## Probability of Matching X out of 6 Numbers, Playing N Numbers

The probability of matching X out of 6 numbers (where X = 4, 5, or 6) by playing N numbers (where N = 6, 7,... 15) is given by the combinatorial formula

[(54 choose N-X) * (6 choose X)] / (60 choose N)

where (A choose B) = A! / [B!(A-B)!]. For example, to compute the probability of matching all 6 numbers when you play only 6 numbers in Mega-Sena you let X = 6 and N = 6. This gives you

[(54 choose 0) * (6 choose 6)] / (60 choose 6)
= 1 / (60 choose 6)
= (6*5*4*3*2*1) / (60*59*58*57*56*55)
= 1 / 50063860
= odds of 1 in 50,063,860
≈ 0.0000000199745

The probability of matching 5/6 numbers when playing 9 numbers in Mega-Sena is

[(54 choose 4) * (6 choose 5)] / (60 choose 9)
= [316251 * 6] / 14783142660
= 459 / 3575990
≈ 0.000128356
≈ odds of 1 in 7,791

The table below summarizes the Mega-Sena lottery odds for all the possible plays and wins.

Odds of Matching 4/6
Odds of Matching 5/6
Odds of Matching 6/6
Overall Odds
1 in 2,332
1 in 154,518
1 in 50,063,860
1 in 2,298
1 in 1,038
1 in 44,981
1 in 7,151,980
1 in 1,014
1 in 539
1 in 17,192
1 in 1,787,995
1 in 523
1 in 312
1 in 7,791
1 in 595,998
1 in 299
1 in 195
1 in 3,973
1 in 238,399
1 in 185
1 in 129
1 in 2,211
1 in 108,363
1 in 122
1 in 90
1 in 1,317
1 in 54,182
1 in 84
1 in 65
1 in 828
1 in 29,175
1 in 60
1 in 48
1 in 544
1 in 16,671
1 in 44
1 in 37
1 in 370
1 in 10,003
1 in 34

## Mega-Sena Prizes

Currently, 35% of Mega-Sena lottery ticket sales are put into the jackpot, 19% is put into the second tier prize for matching 5 out of 6, and 19% is put into the third tier prize for matching 4 out of six. The jackpot is split evenly among winners who have 6 out of 6, and the lower tier prizes are also paid on a pari-mutuel basis -- meaning the prize fund is divided evenly among the winners. Since there is a higher probability of matching 4 out of 6 than 5 out of 6, there are more winners for the third tier prize and thus the payout is less than the second tier prize.

The rest of the income from lottery ticket sales is distributed among special Mega-Sena games. 22% of it funds a drawing held after every five regular Mega-Sena drawings, and 5% of it funds a very special end-of-year drawing offering a huge jackpot.

## Is It Worth It to Buy More Than 6 Numbers per Board in Mega-Sena?

When you play Mega-Sena, your overall odds of winning any prize do not increase in proportion to the price increase of buying more than 6 numbers for a board. Therefore, you have a lower expected return on investment when you buy more than the basic 6 numbers. If you have money to burn, it is better to buy a separate lotto ticket with 6 completely different numbers.

For example, the 7-number board is 7 times as expensive as the 6-number board. But your overall odds of winning any prize with a 7-number board is only about 2.3 times better, not 7 times better.

The 8-number board is 28 times more expensive than the 6 number board, but the overall odds of winning are only about 4.4 times better. The more numbers you buy, the worse the value in terms of overall odds.

However, the jackpot odds do improve proportionally with the price increase.

## Popular

0 of 8192 characters used