California Super Lotto Writeup
Problem Statement
The Rules of the California Super Lotto are:
1) Choose any 5 numbers between 1-47 (these numbers will not repeat).
2) Pick a mega number between 1-27 (this number may repeat).
3) Match all 5 numbers and the mega number and you win!
The questions we are trying to solve are :
1) How many different number combinations are possible for a CA Super Lotto ticket?
2) What is the probability of winning the CA Super Lotto ?
3) If you match all 6 numbers, you win $8,000,000. It costs $1 to play. What are your expected winnings ?
1) Choose any 5 numbers between 1-47 (these numbers will not repeat).
2) Pick a mega number between 1-27 (this number may repeat).
3) Match all 5 numbers and the mega number and you win!
The questions we are trying to solve are :
1) How many different number combinations are possible for a CA Super Lotto ticket?
2) What is the probability of winning the CA Super Lotto ?
3) If you match all 6 numbers, you win $8,000,000. It costs $1 to play. What are your expected winnings ?
Process & Solution
To initially kick off the unit we had "played" the California Super Lotto. No one in the class won. My initial guess for the probability of winning was 1 in a million but it turns out I was way off... so that's how the probability unit started off.
Throughout the entire unit we learned about:
-Probability: A value between 0 and 1 assigned to a random outcome
-Sample Space: Set of all possible outcomes (Ex: rolling a die, sample space=6)
-Event: A successful outcome (Ex: rolling a 6=1)
-P(A): Probability of A (Probability of event A occurring)
P(A)= #of successful outcomes (events)/total outcomes (sample space)
Ex: Rolling a 6/ Rolling a die= 1/6
We also looked at the different types of probability:
Basic probability= P(a)
Complements= P(Ac)
Intersections= P(A and B)
Unions= P(AUB)
After about all of the different probabilities we learned about Dependent vs. Independent events, Conditional probability, and Expected Value. Finally, we had finished learning about everything to do with probability. With everything we had learned it was time to put the skills to the test.
Throughout the entire unit we learned about:
-Probability: A value between 0 and 1 assigned to a random outcome
-Sample Space: Set of all possible outcomes (Ex: rolling a die, sample space=6)
-Event: A successful outcome (Ex: rolling a 6=1)
-P(A): Probability of A (Probability of event A occurring)
P(A)= #of successful outcomes (events)/total outcomes (sample space)
Ex: Rolling a 6/ Rolling a die= 1/6
We also looked at the different types of probability:
Basic probability= P(a)
Complements= P(Ac)
Intersections= P(A and B)
Unions= P(AUB)
After about all of the different probabilities we learned about Dependent vs. Independent events, Conditional probability, and Expected Value. Finally, we had finished learning about everything to do with probability. With everything we had learned it was time to put the skills to the test.
Question #1: How many different number combinations are possible for a CA Super Lotto ticket?
Question #2: What is the probability of winning the CA Super Lotto ?Finding out the total combinations helped a lot with figuring out this problem because all you had to do was divide 120 by the total combination so 120/4,969.962,360 which simplified to 1/4,969,962,360 which means that the probability is that only one person out of the 4,969,962,360 people playing the super lotto will win.
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To begin we started to think of all of the many things we had learned through out this unit and what could help us figure out the total number of possibilities and so we thought back to "Mr.B's Closet" which is a problem we had done that basically has the same concept as the CA Super Lotto problem. What we did for "Mr.B's Closet" was multiply all of the givens (socks, shirts, ties, shoes etc.) together and that'll give us our total combinations, which is exactly what we did for the CA.S.L problem but we took away a number from the sample space because the number cannot repeat so you take one away. The image to the left is correct but then Mr. Carter brought it to our attention that you have any combination of 5 numbers so once you choose one the numerator gets smaller because then you have to choose 4 new numbers and then 3 and so on and so forth. For the mega number you just leave it as 1/27 because you're only choosing it once. Everything multiplied together is 4,969,962,360 its quite a number right well that is the total combinations for a California Super Lotto Ticket.
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Questions #3:If you match all 6 numbers, you win $8,000,000. It costs $1 to play. What are your expected winnings ?
To find out the find out the expected earnings we plugged everything into the expected value formula (shown to your right). Once you solve you should get -0.8, but that's not what we got originally, we had first gotten was -0.9 which is definitely not correct and so we revised our original work about found out the we had put $8,000 instead of $8,000,000 as the payout so once we fixed that we got the -0.8. For every dollar someone spends on the CA.S.L, then the lottery wins 80 cents.
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problem Evaluation
I actually really enjoyed this problem because it made me think back to all of the different probabilities we learned and decide which one would be most helpful to solve this equation. This problem also made me think about who horrible gambling is and how you're not exactly winning anything and the casino or company is actually the one winning. I feel like I definitely got a lot out of the last problem because it had me making many calculations and trying to remember what to next in solving the equation and it just all in all got me thinking hard.
Self Evaluation
I think I should get an A+ for this unit because it really pushed my mind to think on it's own rather then depending on the rest of the group and it honestly helped me a lot more with being independent in math. Being independent was one of my goals for the semester and its made me feel way smarter then I already am and it gave me a good feeling of knowing what was always going on and also being right, that made me feel great as well. But I need to work on speaking up because a lot of the times I would be working on the problem on my own and then the rest of the group would be doing something else that looks completely different from what I was working on but I wouldn't speak up about it and then they would all get the problem wrong.
EDITS
-Add Captions
-Clearer Explanations
-Examples of Work
-Make All Text Match
-Clearer Explanations
-Examples of Work
-Make All Text Match