Bonoloto Odds

Consiste en elegir 6 números del 1 al 49 y a la hora de sellar el boleto, el terminal nos da un número del 0 al 9 paradeterminar el reintegro.
Los sorteos de la Bonoloto se realizan los lunes, martes, miércoles y viernes y el precio de cada apuesta es de 50 Céntimos, siendo el mínimo de apuestas de dos apuestas por boleto.
The following is the probability of winning a prize with 6 correct answers, 5+C, 5, 4 and 3 correct answers:

Category	Hits	Favorable	Odds
1º	6	1/13.983.816	0,0000000715
2º	5 + C	6/13.983.816	0,000000429
3º	5	252/13.983.816	0,0000180
4º	4	13545/13.983.816	0,000969
5º	3	246.820/13.983.816	0,0176

The possible combinations in the game are 13,983,816. That is, there are almost fourteen million possible Winning Combinations arising from the many combinations of 6 numbers out of 49 possible numbers.

The position of each number is not relevant in our probability study, therefore, the order in which the numbers come out is irrelevant.

Therefore, the odds of winning a first category prize with a single bet is one in 13,983,816. For the rest of the categories, the odds of winning a prize with a single bet is one in 13,983,816:

Hits	Probability of success with 1 bet
6 Hits	1 / 13.983.816
5+C Hits	1 / 2.330.636
5 Hits	1 / 55.491
4 Hits	1 / 1.032 (0,097%)
3 Hits	1 / 57 (1,77%)
Refund	1 / 10 (10%) (reimbursement)

The probability of not guessing any number is almost 50%, resulting from dividing the favorable cases (6,096,454) by the possible cases (13,983,816). In the draw there are 6 numbers of the Winning Combination that are the chosen ones, and at the same time there are 43 (49-6) numbers not chosen and therefore, the probability that the six chosen numbers are not the chosen ones is 6,096,454/13,983,816 = 0.436

To secure the first category prize we should invest the amount of 13,983,816/0.50 = 6,991,908 euros. This leads us to consider investing such amount of money if the jackpot is less than this amount, and even in such an assumption, our investment would be ruinous if there is more than one first category winning ticket.

Bonoloto: 6 Lottery Hits

To win the maximum prize with a single bet you have to match all 6 numbers, let's see how likely this is to happen:

• In the first number extracted, six cases out of the possible forty-nine are favorable, so we must divide 6/49 = 0.1224.
• In the second extraction, since one number has already come out, there are five favorable and forty-eight possible numbers left, that is to say 5/48 = 0.1042, and so on, in progression until we reach the sixth extraction, where we would divide the last number by the 44 possible numbers that would be left.
• 4/47 = 0,0851
• 3/46 = 0,0652
• 2/45 = 0,0444
• 1/44 = 0,02273
• 3/46 = 0,0652
• 2/45 = 0,0444
• 1/44 = 0,02273

If you look at these numbers, with each hit the probability that you will hit the next number goes down. If we put these probabilities together we can calculate the probability that we will hit all 6 numbers:
6/49*5/48*4/47*3/46*2/45*1/44 = 720/10.068.347.520 = 1/13.983.816 = 0,00000007

The game consists of guessing 6 numbers out of 49 possible numbers on the ticket. The ticket is a combination of six elements of the possible combinations that can be formed with the numbers 1, 2, 3, 4,..., 49:

C (49 6) = 49*48*47*46*45*44/6! = 13.983.816

Bonoloto: 5 Hits plus complementary

The 2nd category prize is awarded when 5 of the 6 numbers of the winning combination are matched plus the so-called complementary number, which is drawn at random from among the 43 numbers that are not part of the winning combination.

If you have matched 5 of the 6 numbers you have a chance to win a bigger prize. A seventh ball is drawn and if you also match this complementary number, the prize is higher than the 5 correct numbers but less than the 6 correct numbers.
5 out of 6 balls drawn must be missed. Any of the 6 balls chosen can be the wrong ball. Imagine that you miss only the last one. Then the probability of hitting the first five balls and the sixth ball being the complementary is:
6/49*5/48*4/47*3/46*2/45*1/44 = 720/10.068.347.520

But any of the 6 balls can be the wrong ball, so the probability of hitting 5 out of 6 and the complementary is:
6* 720/10.068.347.520 = 6/13.983.816 = 0,000000429

In words it means that per bet played the probability of matching 5 numbers and the complementary is 6 times the probability of matching all 6 numbers, less than 1 in 2 million (2,330,636) chances.

Bonoloto: 5 Hits

The 3rd prize category consists of matching 5 of the 6 winning numbers.

To hit 5 of 6 balls drawn you must miss one. Any of the 6 balls chosen can be the wrong ball. Imagine that you miss only the last one. Then the probability of hitting the first five balls but missing the sixth one (there are only 42 balls left taking away the 6 winning balls and the complementary one) has the following probability:
9/49*5/48*4/47*3/46*2/45*42/44 = 30.240/10.068.347.520

But any of the 6 balls can be the wrong ball so the probability of hitting 5 out of 6 is:
6* 30.240/10.068.347.520 = 181.440/10.068.347.520 = 252/13.983.816 = 0,000018

Summarizing, the probability of matching 5 out of 6 balls in the bonoloto is 252 times the probability of matching all 6 numbers, plus or minus 1 out of 55,491 possibilities.

Bonoloto: 4 Hits

The 4th prize category consists of matching 4 of the 6 winning numbers.

To hit 4 of 6 balls drawn you must miss two numbers. Then the probability of hitting the first 4 and missing the last two is:
6/49*5/48*4/47*3/46*43/45*42*44 = 650.160/10.068.347.520

But since the order of the numbers that matched and did not match does not matter, there are 15 combinations that give the same result:
15* 650.160/10.068.347.520 = 9.752.400/10.068.347.520 = 13.545/13.983.816 = 0,000969

Summarizing, the probability of matching 4 out of 6 balls in the bonoloto is 13545 times the probability of matching the 6 numbers, that is, more or less 1 in 1032 possibilities.

Bonoloto: 3 Hits

The 5th prize category consists of matching 3 of the 6 winning numbers.

To hit 3 of 6 balls drawn you must miss three numbers. Then the probability that you hit the first 3 and miss the last three is:
6/49*5/48*4/47*43/46*42/45*41/44 = 8.885.520/10.068.347.520

But since the order of the correct and incorrect numbers does not matter, there are 20 combinations that give the same result:
20* 8.885.520/10.068.347.520 = 177.710.400/10.068.347.520 = 246.820/13.983.816 = 0,0177

Since 3 of the 6 winning numbers must be matched, the number of possible tickets are combinations of 6 elements taken from 3 in 3, C6,3 by combinations of 43 elements taken from 3 in 3, C 43,3 (once I have matched 3 numbers, the other three can be any of the 43 non-winning numbers).

Summarizing, the probability of matching 3 out of 6 balls in La Bonoloto is 246,820 times the probability of matching all 6 numbers, that is, more or less 1 out of 57 possibilities.

Bonoloto: 0 Hits

Matching any of the 6 numbers is not difficult.

There are 6 good numbers and 43 unwanted numbers, therefore the probability that the six numbers drawn are not good is:
43/49*42/48*41/47*40/46*39/45*38/44 = 4.389.446.880/10.068.347.520 = 6.096.454/13.983.816 = 0,436

In conclusion, the probability of not hitting any number is plus or minus 1 in 2.29