## California Super Lotto Problem**

**The Problem**To focus on probability this unit we used the CA Super Lotto Problem. In this problem we studied the California Super Lotto and the probability of winning.

We were asked these three main questions:

We began by understanding how this lottery works.

We were asked these three main questions:

- How many different number combinations are possible for a CA Super Lotto ticket?
- What is the probability of winning the CA Super Lotto?
- If you match all 6 numbers, you win $8,000,000. It costs $1 to play. What are your expected winnings?\

__The Process__We began by understanding how this lottery works.

- You can choose any 5 numbers between 1-47 and cannot repeat.
- Then, you pick 1 number between 1-27.
- Matching numbers you chose to the ones drawn will result in a win.

**1.) How many different combinations are possible for the CA Super Lotto?**

I started this problem by finding the total possibilities for each individual number. In my initial attempt I thought that there were 47 total combinations for each box because you could pick any number 1 - 47. My initial attempt looked like this:

After discussion I realized that this was wrong because the rules of the lottery state that the first five boxes cannot have repeating numbers. Therefore, for every box the total possible choices would go down one. The first box having 47 choices, the second having 46, and so on. Then, for the mega number, I found that since this number could repeat, it was not affected by the other numbers. The total possible choices for the mega numbers was 27. Lastly, I multiplied them all together. My work looked like this:

**There are 4,969,962,360 total combinations on a CA Super Lotto ticket.**This represents the sample space for the probability of winning. I know that this solution is correct because this problem asked us to find compound probability. When looking for compound probability you are looking for the total probability of multiple events that are dependent upon each other. In this case, the multiple events were the choices of numbers for each individual box and they were dependent on each other because they could not repeat. The possible choices changed based on the number that was previously chosen. The total probability was the total possible combinations.

**2.) What is the probability of winning the CA Super Lotto?**

When starting this problem I knew I had to find compound probability again. This time I found the probability of each chosen number being called in the drawing. Again I knew that the numbers were dependent upon each other so the sample space for each box would go down my by one. My first attempt looked like this:

I thought that the probability of each number being drawn was 1 out of the sample space, which was the possible numbers you could choose from. But I found that this was wrong because in this lottery the numbers called do not have to be in the same order that they are called. In other words, no matter what order the numbers are called, if your numbers match you will win. Therefore, in the first box there are 5 out of 47 chances for the number to be called. In the second box the chances are 4 out of 46 and so on. I found the probabilities of each number being called and multiplied them together. My work looked like this:

**The probability of winning the CA Super Lotto is 0.000000024%**

I know this solution is correct because when multiplying the probabilities you get 120 out of 4,969,962,360 and by dividing that you get 0.000000024%.

**3.) If you match all 6 numbers, you win $8,000,000. It costs $1 to play. What are your expected winnings?**

To start this problem I knew we had to find expected value. Since we just learned how to apply it my first attempt was pretty spot on.

I multiplied the payout by the probability of winning. Then I multiplied the money you could lose playing. Lastly I subtracted the two and simplified it.

**The expected earning is -0.80.**

I know this solution is correct because through the formula I explained above, I was able to find that because my expected earning was a negative, it meant that my odds of winning were so low that the lottery company could expect to make 80 cents for every ticket I bought.

**Problem Evaluation**Throughout this unit I did enjoy this problem. I think that by the end of the unit I was prepared with the tools I needed to complete it. Something that continues to push my thinking despite already having skills in probability is the need to continuously think critically. I saw it especially in the first question where I had to be careful and remember that the sample spaces were different because the numbers were dependent. I think that keeping a critical eye is an essential skill I got to develop further through this problem.

**Self Evaluation**For this unit I would grade myself an A. I worked well in my group being willing to clarify to others when needed and also ask questions that deepened my understanding. I think that a lot of this unit built off of previous lessons and despite the times that I was absent I always kept up with my work and actively participated in class.

**Edits**- Add initial guess for the probability of winning
- Explain how the total combinations represent the sample space
- Make the answers clearer

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