In order to play in the California Super Lotto, you must choose six numbers: 5 of them will not repeat and must be between 1-47, the 6th is your mega number and must be from 1/27, it can be the same as one of your previous numbers. If you get all five of the first numbers correct, regardless of order, plus the mega number, you will win $8,000,000.
While $8,000,000 may seem like a lot of money for a $1 bet, the chances of winning are notorious for being very, very slim. We need to know three things about the CA Super Lotto:
- How many different number combinations are possible for a ticket? - What is the probability of winning? - What are your expected winnings/How much money can the lotto company expect to make for every ticket?
Before Solving
My initial guess was that the probability of winning was 1/1,000,000, but I had an elementary level background on how to solve this problem at the time. After studying probability in class, I now understand that there are a few basic forms of probability that I may encounter.
Key Terms:
Sample space - The total amount you are choosing from. Think, "a bag of variously colored socks and patterned socks". This will be your denominator in your probability.
Event - How many of the item you want to pull out. Think, "5 blue socks". This will be your numerator in your probability.
1) P(A) This is basic probability. Think, "What are the chances I will pull a blue sock out of this bag of variously colored socks and patterned socks?"
2) P(A)^C This would be the opposite of number 1. Think, "What is the probability that I don't pull out a blue sock out of this bag of variously colored socks and patterned socks?"
3) P(A∩B) This is called an intersection. Think, "What is the probability that I pull out a blue sock that also has stripes in this bag of variously colored and patterned socks?" 4) P(A∪B) This is called a union. Think, "What is the probability that I pull out a blue sock or a striped sock in this bag of variously colored and patterned socks?"
5) P(2nd|1st) This is called conditional probability. Think, "What is the probability that I pull out a blue sock given that the sock is striped in this bag of variously colored and patterned socks?" Conditional probability can be used to show that an subgroup is independent of the total.
6) Independent event Independent events are events that do not rely on each other. Think, ""What is the probability that I pull out a blue sock, then put the sock back in this bag of variously colored and patterned socks and pulled a red sock?" Because you put the blue sock back in, the red sock's probability will not be different whether or not you pulled that blue sock first.
7) Dependent event Dependent events are events that do rely on each other. Think, ""What is the probability that I pull out a blue sock, then don't put the sock back in this bag of variously colored and patterned socks and pulled a red sock?" Because you didn't put the blue sock back in, the red sock's probability will be different. The sample space is now smaller than before.
8) E(X) Expected value is how much money or value is expected over time based on probability. Think, "If I get $5 every time I pull a blue sock, and it costs $1 to pull a sock, how much money can I expect to win over time?"
Solving the California Super Lotto Problem
Question 1 - How many different number combinations are possible for a ticket?
My group and I first thought that we would have to multiply 47^5 by 27 because each of the first five numbers has to be 1-47 and the last has to be 1-27.
We then realized that this was wrong because the first five numbers cannot repeat. This made the event dependent and the sample sizes would get smaller with each number. Here was our final sample space:
Question 2 - What is the probability of winning?
After finding the sample space, we still had to find the event. Similar to before, we treated the events as independent and wrote 5/47, 5/46, 5/45, 5/44, 5/43, 5/42, and 1/27. This was incorrect because although we did account for the fact that the first 5 numbers could be in any order by putting 5 as the numerator, we forgot again that the numbers could not repeat. So we changed it to this and found the answer for our probability:
Although my original guess of 1/1,000,000 was wrong, now that I understand the math behind it, this answer makes much more sense.
Question 3 - What are your expected winnings/How much can you expect to lose for every ticket over time?
Calculating for expected value is a little more tricky than the first two questions, so we looked back at our previous notes to solve this problem. In roulette, we had to figure out how much the house could expect to win over time by multiplying the expected winnings to the probability of winning, and the money it costs to play (-1) by the probability of losing and added the two together. This showed us that the money you win on each bet actually is calculated so that the house wins at least $0.05 for every one dollar spent on the game.
The California Super Lotto is set up similarly. We multiplied the $8,000,000 that you'd win by the probability of winning, and the $1 it costs to play (-1) by the probability of losing and added to two together. This was our answer:
Problem Evaluation
I liked this problem and thought it was a good test of all of the things we had learned throughout the unit. Putting expected value to the test was my favorite part because it was more complex than the first two questions, but I thought that all of them had parts that pushed our thinking.
Self Evaluation
If I were to grade myself on this unit, I would give myself an A+. I think I worked really hard this unit and I usually ended up finishing my work early. I especially worked hard to write out my notation on every problem and drew graphs and wrote statistics to prove my points about probability.