Introduction
When you write your 5 numbers and 1 mega number on your Super Lotto ticket you might think, "you never know, I might get lucky." It depends on how you define "lucky." Today I will show you how much of a chance you can become a millionaire. Are these lottery companies ripping you off?
The rules are simple. Choose 5 numbers that may not repeat and 1 mega number that may repeat one of your 5 chosen numbers. Match all 5 numbers and your mega number then you become rich!
Focus Questions:
a)How many different number combinations are possible for a CA Super Lotto ticket?
b) What is the probability of winning the CA Super Lotto?
c) If you match all 6 numbers, you win $8,000,000. It cost $1 to play. What are your expected winnings?
The rules are simple. Choose 5 numbers that may not repeat and 1 mega number that may repeat one of your 5 chosen numbers. Match all 5 numbers and your mega number then you become rich!
Focus Questions:
a)How many different number combinations are possible for a CA Super Lotto ticket?
b) What is the probability of winning the CA Super Lotto?
c) If you match all 6 numbers, you win $8,000,000. It cost $1 to play. What are your expected winnings?
Starting the Problem
We gotta start somewhere right? Our teacher asked us to make a guess. My first reaction was "I don't know anything how the heck am I suppose to make a guess?" I ended up with this uneducated guess...
After we guessed we were left with our table to answer the focus questions. With no direction whatsoever from Mr.Anatole. We had no clue what to do! We were left there with 10 long seconds of gazing at each other with blank stares. I did what I could do and asked for a marker which was dull. I asked for another one and started writing what we knew.
a) There is a total of 5 numbers each one with 1 less of choice to choose from each time you pick a number
b)5 number choices of numbers 1-47 but can't repeat!
c)1 number choice of numbers 1-27
Everyone agreed. "What do we do next?" Silence. No one replied. We threw some ideas, but didn't make sense and only confused us more.
We needed guidance so we asked the teacher and he showed us the method called Permutations. It is the funniest thing! I did this on Khan! I really disliked it too. What kind of fate is this! I felt like face palming the heck out of myself. Duh! Everything was so clear now!
a) There is a total of 5 numbers each one with 1 less of choice to choose from each time you pick a number
b)5 number choices of numbers 1-47 but can't repeat!
c)1 number choice of numbers 1-27
Everyone agreed. "What do we do next?" Silence. No one replied. We threw some ideas, but didn't make sense and only confused us more.
We needed guidance so we asked the teacher and he showed us the method called Permutations. It is the funniest thing! I did this on Khan! I really disliked it too. What kind of fate is this! I felt like face palming the heck out of myself. Duh! Everything was so clear now!
Video I watched of the wonderful topic to better understand the subject I oh so disliked to do.
Is this...Fate?
What I wrote with my group after finally having a direction to answer the focus questions
do You may be asking okay I don't get what's funny what is Permutations Kayla...Glad you asked! I am going to try my absolute best to explain to you because I never understood it the first 3 days I was introduced to this method.
Lets us do a simple Khan Academy problem:
How many times can you arrange the word SONG?
You have 4 choices of spots to arrange the letter S
___ ___ ___ ___
4 x
S ___ ___ ___
You have 3 choices of spots to arrange the letter O
S O ___ ___
4 x 3
You have 2 spots to arrange the letter N
S O N ___
4 x 3 x 2
You have 1 spot to arrange the letter G
S O N G
4 x 3 x 2 x 1= 24 total times you can arrange the word SONG
Lets us do a simple Khan Academy problem:
How many times can you arrange the word SONG?
You have 4 choices of spots to arrange the letter S
___ ___ ___ ___
4 x
S ___ ___ ___
You have 3 choices of spots to arrange the letter O
S O ___ ___
4 x 3
You have 2 spots to arrange the letter N
S O N ___
4 x 3 x 2
You have 1 spot to arrange the letter G
S O N G
4 x 3 x 2 x 1= 24 total times you can arrange the word SONG
But..Wait! There's more! What if...
Dig this...4 x 3 x 2 x 1 is the same as saying 4! This means it multiplies every number till it reaches the number 1.
But Kayla, what if you don't have the same amount of spots to arrange each letter in?
Lets say...
You have 5 horses and you only have enough stalls to fill 3 horses. How many times can you arrange each horse?
You do the same symbol ! and use division to cancel out the spots you don't have,for example...
5 x 4 x 3 x 2 x 1= 5!
2 x 1= 2!
Answer: 5!/2! or (5 x 4 x 3)= 60 ways to arrange 5 horses in each of the 3 stalls
But Kayla, what if you don't have the same amount of spots to arrange each letter in?
Lets say...
You have 5 horses and you only have enough stalls to fill 3 horses. How many times can you arrange each horse?
You do the same symbol ! and use division to cancel out the spots you don't have,for example...
5 x 4 x 3 x 2 x 1= 5!
2 x 1= 2!
Answer: 5!/2! or (5 x 4 x 3)= 60 ways to arrange 5 horses in each of the 3 stalls
Using Permutations to answer some questions
a)How many different number combinations are possible for a CA Super Lotto ticket?
47 choices
You pick one number
47 x 46
Picking your second choice (down to 46 choices of numbers since number can't repeat)
47 x 46 x 45
Picking your third choice(down to 45 choices of numbers since number can't repeat)
47 x 46 x 45 x 44
Picking your fourth choice (down to 44 choices of numbers since number can't repeat)
47 x 46 x 45 x 44 x 43
Picking your fifth choice (down to 43 choices of numbers since number can't repeat)
How about the Mega number? You have a total of 27 numbers to choose from and you don't have to worry about repeating:
47 x 46 x 45 x 44 x 43 x 27= 4,969,962,360 different combinations
(47!/42!) x 27= 4,969,962,360
b) What is the probability of winning the CA Super Lotto?
We know that there are 4,969,962,360 total combinations. When you pick you 5 numbers and mega number you have to get the same numbers but not in the same order. How many total combinations of chance can you get based on the numbers you have chosen? What is the probability of you winning the CA Super Lotto?
Lets use the Permutation method..again!
P (of winning)= ___ ___ ___ ___ ___ ___
Numerator:You have 5 total spaces to put your chosen number
Denominator: 47 total numbers to choose from
P (of winning)= 5/47 x ___ ___ ___ ___ ___
Numerator:You have 4 total spaces to put your chosen number
Denominator: 46 total numbers to choose from
P (of winning)= 5/47 x 4/46 x ___ ___ ___ ___
Numerator:You have 3 total spaces to put your chosen number
Denominator: 45 total numbers to choose from
P (of winning)= 5/47 x 4/46 x 3/45 x ___ ___ ___
Numerator:You have 2 total spaces to put your chosen number
Denominator: 44 total numbers to choose from
P (of winning)= 5/47 x 4/46 x 3/45 x 2/44 x ___ ___
Numerator:You have 1 total spaces to put your chosen number
Denominator: 43 total numbers to choose from
P (of winning)= 5/47 x 4/46 x 3/45 x 2/44 x 1/43 ___
Mega Number: Numerator:You have 1 total spaces to put your chosen number
Denominator: 27 total numbers to choose from
P (of winning)= 5/47 x 4/46 x 3/45 x 2/44 x 1/43 x 1/27= 120/4,969,962,360 probability of winning the CA Super Lotto
c) If you match all 6 numbers, you win $8,000,000. It cost $1 to play. What are your expected winnings on average?
You know that you make a total of 8,000,000 dollars of you win the Lottery
You lose 1 dollar per ticket if you lose the Lottery
Well how the heck do I start this problem? Mr. Anatole was a wonderful person and gave us a formula to start with:
8,000,000 (P winning) + (-1) (P losing)
This was the formula he gave us and it was up to me to insert the probability of winning and losing.
I knew that you had a 120/4,969,962,360 chance of winning. How about losing? I subtracted 120 from 4,969,962,360 and ended up with this number: 4,969,962,240
I knew that we had a 4,969,962,360 total combinations. You had a 4,969,962,240/4,969,962,360 chance of losing. The odds are totally against me.
I ended up with this equation:
8,000,000 (120/4,969,962,360) + (-1) (4,969,962,240/4,969,962,360)= -0.80
-.80 cents is your total expected winnings.
47 choices
You pick one number
47 x 46
Picking your second choice (down to 46 choices of numbers since number can't repeat)
47 x 46 x 45
Picking your third choice(down to 45 choices of numbers since number can't repeat)
47 x 46 x 45 x 44
Picking your fourth choice (down to 44 choices of numbers since number can't repeat)
47 x 46 x 45 x 44 x 43
Picking your fifth choice (down to 43 choices of numbers since number can't repeat)
How about the Mega number? You have a total of 27 numbers to choose from and you don't have to worry about repeating:
47 x 46 x 45 x 44 x 43 x 27= 4,969,962,360 different combinations
(47!/42!) x 27= 4,969,962,360
b) What is the probability of winning the CA Super Lotto?
We know that there are 4,969,962,360 total combinations. When you pick you 5 numbers and mega number you have to get the same numbers but not in the same order. How many total combinations of chance can you get based on the numbers you have chosen? What is the probability of you winning the CA Super Lotto?
Lets use the Permutation method..again!
P (of winning)= ___ ___ ___ ___ ___ ___
Numerator:You have 5 total spaces to put your chosen number
Denominator: 47 total numbers to choose from
P (of winning)= 5/47 x ___ ___ ___ ___ ___
Numerator:You have 4 total spaces to put your chosen number
Denominator: 46 total numbers to choose from
P (of winning)= 5/47 x 4/46 x ___ ___ ___ ___
Numerator:You have 3 total spaces to put your chosen number
Denominator: 45 total numbers to choose from
P (of winning)= 5/47 x 4/46 x 3/45 x ___ ___ ___
Numerator:You have 2 total spaces to put your chosen number
Denominator: 44 total numbers to choose from
P (of winning)= 5/47 x 4/46 x 3/45 x 2/44 x ___ ___
Numerator:You have 1 total spaces to put your chosen number
Denominator: 43 total numbers to choose from
P (of winning)= 5/47 x 4/46 x 3/45 x 2/44 x 1/43 ___
Mega Number: Numerator:You have 1 total spaces to put your chosen number
Denominator: 27 total numbers to choose from
P (of winning)= 5/47 x 4/46 x 3/45 x 2/44 x 1/43 x 1/27= 120/4,969,962,360 probability of winning the CA Super Lotto
c) If you match all 6 numbers, you win $8,000,000. It cost $1 to play. What are your expected winnings on average?
You know that you make a total of 8,000,000 dollars of you win the Lottery
You lose 1 dollar per ticket if you lose the Lottery
Well how the heck do I start this problem? Mr. Anatole was a wonderful person and gave us a formula to start with:
8,000,000 (P winning) + (-1) (P losing)
This was the formula he gave us and it was up to me to insert the probability of winning and losing.
I knew that you had a 120/4,969,962,360 chance of winning. How about losing? I subtracted 120 from 4,969,962,360 and ended up with this number: 4,969,962,240
I knew that we had a 4,969,962,360 total combinations. You had a 4,969,962,240/4,969,962,360 chance of losing. The odds are totally against me.
I ended up with this equation:
8,000,000 (120/4,969,962,360) + (-1) (4,969,962,240/4,969,962,360)= -0.80
-.80 cents is your total expected winnings.
A classmates interpretation of solving this problem. She helped me a lot and I am pretty grateful
Self Evaluation & Problem Evaluation: Whole Week & 6 Week Probability Unit
Did I like this problem? Nope, I disliked it whole hearty. I've always sucked at fractions and had dark times with them. I also dislike probability since I make it more complicated than it really is. It did push my thinking since I didn't want to fall behind. I was confused for most of this unit and I even did Khan for weeks on probability! I just couldn't wrap my head around it. I am positive that I improved much on probability so that good right? This California Super Lotto problem came easy for me and I didn't really need as much help as I usually do. I was the one who took the lead in this lottery problem with my group. I have seen good results on my understanding.
If I had to receive a grade on the whole 6 weeks I would give myself an A because I really tried. I am not the best person at understanding this unit, but I have seen so much growth in myself. I did get of task, but that was because I didn't understand anything no matter how much I asked my table mates or teacher. I tried to be independent more than dependent which trained me to do this on my own. I didn't do well on the test at first and did test corrects. You know what? I reviewed it and I knew exactly what I did wrong. I am just slow in the skull that's all. It takes patience. |
TuesdayStarting this problem wasn't easy since half the class tried to get into the right direction to actual solve one of the focus questions. We didn't really get off task though, so that's good. I would give myself a 5 that day since I came up with two answers for the focus question.
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WednesdayToday I was productive. I got enough done to not be behind I would give myself a 4 today.
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ThursdayToday I was getting off task, but I got more done than any other day. I give myself a 4 because I worked hard.
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FridayI would give myself a 4 because I barely got sleep last night because of this write up. I need to reward myself.
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