Also remember the format for submission. If you do not follow the format, your solution will be posted but I have no way to give you credit. example - JL per. 3
Due Nov 8th
Solution:
Most people use a probability tree to start sorting out their combination totals for each item. Here is an example.
Also, I decided to give an accurate solution put together by a Middle School Student from California. Their method to use x! is a probability technique that is practiced in middle school across the country - state standards.
From:
Maya George, age 12
School: Howell Township Middle School North, Howell, NJ
There are 716,636,160 possible combinations.
The odds that both Kaylee's and Kelly's names will be first in the different combinations is 1 to 17.
To solve the first question of this problem, I took the problem step by step. Since there are 4 students running for the position of president, I started with using the letters "A,B,C,D" to represent each student. Then I listed the possible combinations:
ABCD BACD CABD DABC
ABDC BADC CADB DACB
ACBD BCAD CBAD DBAC
ACDB BCDA CBDA DBCA
ADBC BDCA CDBA DCAB
ADCB BDAC CDAB DCBA
This made a total possible of 24 arrangements for the president
position alone.
There are a total of 3 students running for the position for vice-
president. So I used the letters "A,B,C" to represent each student. I
then listed all the possible combinations:
ABC BAC CAB
ACB BCA CBA
This made a total of 6 possible arrangements for the vice president
position alone. Now I looked at the two lists I had made. The first
one had 4 people running for president, and resulted in 24 possible
combinations. The second had 3 people running for vice-president, and
resulted in 6 possible combinations.
After thinking for awhile, I realized that if there were "x" people running for the position, the
formula for finding the number of combinations would be x*(x-1)*(x-2)* (x-3)...1 For example, in the first set of arrangements, there were 4 people running for president and the number of combinations was 4*3*2*1=24. In the second set of arrangements, there were 3 people running for vice-president, and the number of combinations was 3*2*1=6.
Using this formula, I came up with a table. This is how it looked:
POSITION | NUMBER OF COMBINATIONS
------------------------------------------------------
PRESIDENT| 4*3*2*1= 24
VICE-PRESIDENT| 3*2*1= 6
SECRETARY| 4*3*2*1= 24
TREASURER| 2*1=2
8TH GRADE REPRESENTITIVE| 3*2*1=6
7TH GRADE REPRESENTITIVE| 4*3*2*1= 24
6TH GRADE REPRESENTITIVE| 6*5*4*3*2*1=720
Now, to find the TOTAL number of possible combinations, I multiplied the number of combinations by each other. So I multiplied 24*6*24*2*6*24*720. I got a resulting product of 716,636,160. So this was then my final answer for the first question.
To find the odds that both Kaylee and Kelly's name will be first on the list of representatives, I first had to find the odds that each of their names will be first on this list, and I would then multiply them together. To do this, I took Kelly first. He was one of the 6 people running for 6th grade representitive. So he had a one out of six chance of being first on the list. This means that the probability that his name will be on top of the list is 1/6.
Kaylee was one of the three people running for 8th grade representative. So she had a one out of three chance of being first on the list, and so the probability of her name being first on the list would be one out of three, or 1/3.
Now I multiplied these together to find the total probability that their names will be on top. 1/3*1/6= 1/18, so 1/18 is the total PROBABILITY that their names will be on top. Now I needed to convert this to the odds.
1/18 stands for the number of total combinations that Kelly and Kaylee are BOTH listed first as a fraction of the total number of all possible combinations. If the number of combinations that their names are listed first is 1/18, 17/18 is the number of arrangements when their names aren't listed first. Since the definition of ODDS is the number of chances for an event to happen versus the number of chances against the event happening, the ODDS
that they both are listed first in their respective races is 1 to 17.
AA period: 3
ReplyDeleteP (Kelly first for 10th grade)= (1/4)
12th Grade Election:
There are 3 students running so the probability that Kaylee's name will be listed first is:
P (Kaylee first for 12th grade) = (1/3)
The combined probability that both Kelly and Kaylee will have their names first is:
P (Both first) = P (Kelly first) × P (Kaylee first) = (1/4) × (1/3) = (1/12)
The probability that both Kelly and Kaylee will have their names listed first in their elections is 1 in 12. (I don't know how to turn that into percentage.)
AK per 3 the odds are 1/12
ReplyDeleteJJ per 3. their could be a infinite amount of arrangements, their could be a 2/1000 change of their names being listed
ReplyDelete1 out of 4
ReplyDelete58 possible arrrangements Kelly has a 1 out of a 36 chance and Kaylee has a 1 out of 9 chance
ReplyDelete