Counting Story Performance Task Detective Conan - A Meaningful Day -

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Counting Story Performance TaskDetective Conan-
A Meaningful Day -

• MDM4U1-02Hall Lo

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• Conan today goes to school as usual. Today the
  teacher wants to form a committee in class which
  is responsible for collecting homework. The
  committee should consist of 5 students.
• To be fair, the teacher decided to choose the
  group randomly as there are 30 students in the
  class, so the teacher prepared 30 small sheets of
  paper having one students name on each of them
  without repetition . Then she will choose 5 of
  the 30 paper out without replacement.

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• Here are Conans 4 friends
• Mitch Amy Ai George What is
  the probability that Conan and his 4 friends will
  be chosen to be the committee at the same time?

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Since there are 30 students in the class, the
total number of choosing 5 students out can be
calculated by using the Combination formula.
nCr The reason why we use combination is
because the choosing order of the students isnt
important.So, the total number of ways to choose
5 students from the 30 students is 30C5
142506. By using the same formula, we can also
calculate the number of choosing Conan and his
four friends. As there
Total number of items can be chosen
n!r! (n r)!
Number of items has to be chosen
are 5 of them and 5 students have to be chosen
from them, the number of ways is 5C5 1.
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From the data from last slide, we can calculate
the probability by applying the Probability
formula. P(A) As stated above, the
sample space includes all possible outcomes. An
event is a group of outcomes, and an outcome is a
possible result that the event can happen. So,
the probability of Conan and his 4 friends are
chosen into the 5-student committee 5C5
1 30C5 142506 Oh they all
are selected as the committee!They are so lucky
today!
Number of outcome in which Event A can occur
n(A) n(S)
Total number of possible outcomes (Sample space)
0.0007017
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• After getting Conan and his 4 friends as the
  committee, the teacher wants to choose their
  positions. They only have 5 subjects which have
  homework. The 5 subjects are English,
  Mathematics, History, Music and Science.
• Conan wants to be the Mathematics homework
  collector, while his 4 friends all want to be the
  English homework collector. So the teacher
  decides to choose the positions for them. She
  uses 5 pieces of paper this time, on each of them
  there is one of the 5 students name. The teacher
  will choose one piece of paper for the
  Mathematics homework collector first, and then
  choose other pieces of paper for other subjects
  homework collectors, without replacement.

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What is the probability that Conan will be
chosen as the Mathematics homework collector
while his 4 friends can be any of the other
subjects homework collector?
I want to be the Mathematics homework collector
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• As Conan must be the Mathematics homework
  collector, that means there is only 1 choice for
  the Mathematics homework collector. After that,
  we can apply the Permutation formula to calculate
  number of ways of arranging the remaining 4
  friends.
• nPr
• As there are 4 friends that have to be arranged
  for the remaining 4 subjects homework
  collectors, so we use 4P4(4 x 3 x 2 x 1) for the
  number of arrangements for them. So the total
  number of arrangements is 1 x 4P4. We use
  multiplication because these two events has to
  happen at the same time. This is also called
  multiplicative counting principle.

Total number of items
n! (n r)!
Number of items has to be taken at one time
1 x 4 x 3 x
2 x 1 no. of arrangements for
number of arrangements for the rest toone
person to be chosen be chosen for the
remaining 4 subjects for the math homework
homework collectorscollector
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SAMPLE SPACE
Mathematics
Music
Science
English
History
4
3
2
1
1
Number of arrangements 1 x 4 x 3 x 2 x
1 24
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• For the total number of arrangements, we can
  apply the Permutation formula again. As there are
  5 students (Conan and his friends) have to be
  arranged for the different positions, total
  number of ways will be 5P5, which will be 5!
  120
• From the previous slide, we know that there are
  24 arrangements for Conan being the Mathematics
  homework collector. So, we can calculate the
  probability by using the Probability formula
  again.

1 5
24 120
Probability

• Conan really is selected as the Mathematics
  homework collector! He is so lucky!

Unbelievable.. I am so lucky today!Am I
Dreaming?
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• After few lessons, it is now lunch time. The
  students have 3 choices for their lunch Set A,
  Set B or Set C.
• Set A Salad chocolate milk
• Set B Bread egg sausages
• chocolate milk
• Set C Sandwich grilled Fish soup
• The lunch is randomly given to the students.

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Conan doesnt want to have Set A as his lunch as
he thinks that it is strange to have salad and
chocolate milk at the same time But he likes Set
B or Set C. What is the probability that he
will not get Set A as his lunch?
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We can get the answer by getting the probability
of getting Set A as lunch first. After that, as
the probability of NOT getting Set A as lunch is
a complement of that of getting Set A as lunch,
we can simply use 1 to minus the probability of
getting Set A as lunch. This is known as a
formula also. Probability of getting Set A
as lunch 1/3So, probability of NOT
getting Set A as lunch 1 1/3 2/3
Outcomes make up A
P(A) P(A) 1
Outcomes make up A (event A doesnt happen)
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Oh, Conan is still lucky as he gets Set C for
his lunch. He is happy at first, but then he
discovers that the grilled fish doesnt taste
good. So he finishes his lunch unhappily.
Tastes bad (
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• Time flies..And they enjoy the visit of the
  Museum. They are so hardworking that they decided
  to go to the library and do the project together.

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• There are 4 parts in the project Introduction
  part, Information part, Q A part, Conclusion
  part. They decide that every person is
  responsible for one part, and after that they
  will discuss and combine the parts into a
  project. Amy is very good in designing, so she
  just had to design the layout of the project and
  didnt need to do any of the 4 parts.

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• To be fair, they numbered the parts and prepared
  4 papers having numbers 1 4 on each of them.
  Then each of the 4 people will choose 1 of them.
  The numbers refer to the following parts
• 1 ? Introduction2 ? Information3 ? Q A4 ?
  Conclusion
• They will get the papers in this order Conan,
  Ai, George, Mitch. What is the probability that
  Conan will do the Conclusion part and Ai will do
  the Introduction part?

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1 2 3
4 1 2 4
3 1 3 2
4 1 3 4
21 4 2
31 4 3
22 1 3
42 1 4
32 3 1
42 3 4
12 4 1
32 4 3
13 1 2
43 1 4
23 2 1
43 2 4
13 4 1
2 3 4 2
14 1 2
34 1 3
24 2 1
34 2 3
14 3 1
24 3 2
1

• These are dependent events, as the papers Ai can
  get will be affected by what Conan has taken.

SAMPLE SPACE
Probability that Conan will do the Conclusion
part and Ai will do the Introduction part
2/24 1/12
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We can check the answer we just got by using the
Conditional Probability formula. Conditional
probability is the probability that B occurs,
given that A has already occurred. Let event A
be Conan does the Conclusion part.Let event B be
Ai does the Introduction part. By applying the
formula, we can get the P(A and B), which
represents Conan will do the Conclusion part
while Ai will do the Introduction part.P(A)
1/4 (There are 4 parts left and Conan only
want 1 of them)P(BA) 1/3 (There are 3
parts left and Ai only want 1 of
them)So, P(A and B) 1/4 x 1/3 1/12We can
see that we got the correct answer! )
P(BA) x P(A) P(A and B)
Conditional Probability
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• So, what if the question becomes this
• What is the probability that Conan will do the
  introduction part OR Ai will do the introduction
  part?
• For this question, we have to get the
  probability that Conan will do the introduction
  part and the probability that Ai will do the
  introduction part respectively first.
• P(Conan do the introduction part) 1/4P(Ai do
  the introduction part) 1/4
• So, the probability that Conan will do the
  introduction part OR Ai will do the introduction
  part will be 1/4 1/4 1/2. We use additive
  counting principle here because the 2 tasks are
  mutually exclusive. They cannot do the
  introduction part at the same time as each of
  them should be responsible to each different part.

We cannot do the introduction part at the same
time
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• After doing the project for 2 hours, they feel
  hungry. They decided to go to a nearby restaurant
  to have dinner together.

Yeah!Dinner Time!
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• They look at the menu. They find that some of
  the dinner set contains vegetables, some contains
  meat, and some contains egg as following
• - 6 of the dinner set contain meat
• - 8 of the dinner set contain vegetable
• - 7 of the dinner set contain egg
• - 3 of the dinner set contain meat and vegetable
• - 2 of the dinner set contain meat and egg
• - 2 of the dinner set contain vegetable and egg
• - 1 of the dinner set contain meat, vegetable,
  egg
• How many dinner sets are there altogether?

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• We will use a Venn diagram to demonstrate this
  question.

S
Meat
Egg
1
2
4
Subset
1
2
1
6
Vegetable
We find that these are non-mutually exclusive
events. There are some overlapping in the Venn
diagram. Total number of sets
2112416 17
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• After a while, Conan receives his coffee. There
  are 3 bottles of powder 1 containing sugar, 1
  containing salt, and 1 containing milk powder.
  Conan is so lazy that he will just choose any 2
  of them with replacement and add them into the
  coffee and drink it! He will choose 1 and add it
  to the coffee and then put the bottle of powder
  back and choose another 1 and add it to the
  coffee again.
• What is the probability that he will get sugar
  and milk powder?

Sugar Salt Milk Powder
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• Let event A be getting a sugar bottle.Let event
  B be getting a milk powder bottle.
• These are independent events as there is
  replacement. The probability of event A will be
  1/3, as it is to choose 1 of the 3 bottles out.
  The probability of event B is also 1/3, because
  after event A the bottle is put back, and there
  still have 3 bottles for Conan to choose. So the
  probability of event A and event B is 1/3 x 1/3
  1/9.

Wish me luck!
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• Actually, Ai knows about the bottles. She knows
  which of them contains which kind of powder.
  However, she doesnt want to tell Conan directly
  about the correct bottles. She just let him
  choose one of the bottles for 9 times with
  replacement, and then she will tell him how many
  times he gets the bottles correctly. So, Conan
  get one of the bottles with replacement for 9
  times. That means he carries out 9 trials.
• A the bottle containing sugar.B the bottle
  containing salt.C the bottle containing milk
  powder.

ABACCBACA
OUTCOMES
Which one should I choose?
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• From the results of last slide, we can calculate
  the experimental probability.
• The experimental probability that Conan get the
  bottle containing sugar (Bottle A) is 4/9, the
  bottle containing salt (Bottle B) is 2/9, and the
  bottle containing milk powder is 3/9 1/3.
• We discover that the experimental probability is
  different from the theoretical probability. This
  is because of chance variations. The theoretical
  probability will always the same for an event,
  while the experimental probability can be
  different every time for an event.
• With the help of Ai, Conan get the 2 correct
  bottles. He enjoys the coffee.

Thanks Ai
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• After they finished their dinner, they decide to
  go home. It is already evening. They really had a
  meaningful day!

Thanks for watching!
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Credits

• YTV Japanhttp//www.ytv.co.jp/conan/(for the
  screenshots of Detective Conan from the TV
  episodes)
• Xenical http//www.xenical.com.ph/eatingright.htm
  l (for the pictures of food)
• Cloverleaf International http//www.cloverleaf.com
  .hk/fadeProducts2.php (for the picture of sugar
  bottle)
• APTX forumhttp//bbs.aptx.cn/ (sources of
  Detective Conan TV episodes)
• Gosho Aoyama(for he creates Detective Conan)
• Shogakukan(Publisher for Detective Conan)