Swimming Instructor

Subset: elements found in one set are also members of another set

Cynthia is part of the Lifesaving Society. Her job is a swimming instructor who teaches little kids. She teaches 6 classes of different levels. These classes can be called the subsets of who Cynthia teaches. Cynthia is a subset of the Lifesaving Society. 

S = All members of the Lifesaving Society
C = Cynthia (Instructor)

∴ Swimmer 6 ⊂ C
     H4O2 ⊂ C
     Preschool 3 ⊂ C
     Preschool 4 ⊂ C
     Swimmer 5 ⊂ C
     Swimmer 2 ⊂ C
     C ⊂ S

Washing Dishes

Independent Events

Kelvin and Cynthia are the ones in charge of washing dishes from Monday to Friday. The tree diagram list the possible outcomes of Kelvin or Cynthia washing dishes from Monday to Friday. The probability of Kelvin washes dishes is 11 out of 20.

Cynthia washes dishes 9 out of 20. Because Kelvin washes dishes the most, he complains to our parents that Cynthia doesn’t wash the dishes. He calculates for our parents what the probability of Cynthia washing dishes at least three of the five days is.

                                    

Answer: It is an independent event because the probability of an event occurring is not affected by an event that has occurred. The probability of Kelvin and Cynthia washing dishes on a given night is fixed. Therefore, we multiply the probabilities of the five nights to determine the probability of each outcome.

A = Cynthia Washing dishes
P (A>_3) = KKCCC + KCKCC + KCCKC + KCCCK + KCCCC + CKKCC + CKCKC +                    CKCCK  + CKCCC + CCKKC + CCKCK + CCKCC + CCCKK + CCCKC +                                  CCCCK + CCCCC
                = 0.0276 + 0.0276 + 0.0276 + 0.0276 + 0.0226 + 0.0276 + 0.0276 + 0.0276 + 0.0226 +
                   0.0276  + 0.0276 +0.0226+ 0.0276 + 0.0226 + 0.0226 + 0.0184
                = 0.4074 / 40.74%

∴ The probability that Cynthia washes dishes for three or more days in a week is 40.74%.
∴ Kelvin is correct that he washes more dishes than Cynthia.

Down the Road G2

Conditional Probability

Kelvin and Cynthia are getting their G2. Kelvin is the first to take the test and Cynthia takes it next. According to PassTheWheel.com, the pass rate is 64%. The probability of Kelvin and Cynthia passing is 70%. Kelvin failed on his first attempt. Cynthia is nervous for her driver’s test and she calculates the probability of her failing given that Kelvin failed.

Answer: This is a conditional probability problem because the probability of Cynthia failing her G2 (B) is created based on the probability of Kelvin failing his G2 (A). Therefore, the probability of B is dependent of the probability of A.
 

A = Kelvin failing G2                        

B = Cynthia failing G2

P(A∩B) = 1-0.7 = 0.3

P(A) = 1.0.64 = 0.36

P(B|A) = P(A∩B)

                 P(A)

            = 0.3

               0.36

            = 0.83333/ 83.33%

∴ the probability Cynthia failing the test is 83.33% given that Kelvin failed the first time. 

Social Friends

Mutually  & Non Mutually Exclusive Events

Kelvin and Cynthia have their own groups of friends at St. Brother Andre. Some of Kelvin’s friends are friends with Cynthia. Same goes with some of Cynthia’s friends, they are friends with Kelvin. However, Cynthia also has friends from work and Kelvin doesn’t have any work friends because he doesn’t have a job. We can say Cynthia’s work friends are mutually exclusive to Kelvin and his friends because they have never met one another and do not have a connection. Conversely, the friends that are common with both Cynthia and Kelvin are considered to be non-mutually exclusive events. In total Cynthia has 308 friends, 41 from work and Kelvin has 84 friends. Kelvin and Cynthia found that the total number friends they have in common are 33.

Additive Principle:
Answer: The example gives 2 sets (# of Cynthia's friends and # of Kelvin's friends). The number of elements in A∪B can be found by adding the total of number of elements in both sets and subtracting the elements that have been counted twice (A∩B).

Kelvin and Cynthia decided to figure out the number of friends that are friends with Kelvin or with Cynthia.  

                          

A = # of friends Kelvin has

B = # of friends Cynthia has

n (A ∪ B) = n (A) + n(B) – n(A ∩B)

                 = 84 + 308 – 33

                 = 359

 ∴ 359 is the number of friends that are friends with either Kelvin or Cynthia.

Twins Same Food

 

Experimental Probability
Experimental probability is the observed probability of an event in an experiment.
 

We decided to conduct an experiment to see how many favourite foods we have in common. We decided to list out 15 of our favourite foods.

A = # of times we have the same favourite food

P(A) = n(A)

            n(S)

         = 7/15

∴ the probability that both of us liking the same food is 7/15.