REPRESENTING SAMPLE SPACES IN DIFFERENT WAYS

Problem 1 :

Represent the sample space for tossing two coins using 

a) a list

b) 2-D grid

c) a tree diagram

Solution :

In every coin, we see two faces. Head and tail.

Let H be the head and T be the tail.

a) a list

S = {HH, TT, HT, TH}

b) 2-D grid 

c)  tree diagram

Problem 2 :

Illustrate, using tree diagram, the possible outcomes when drawing two marbles from a bag containing several marbles of each of the colors, red green and yellow.

Solution :

Let R be the red marble

G be the green marble and

Y be the yellow marble.

List the sample space for the following.

Problem 3 :

Twirling a square spinner labelled A, B , C and D.

Solution :

Sample space = {A, B, C, D}

Problem 4 :

The sexes of a 2 child family.

Solution :

Let B be the boy child and G be the girl child.

Sample space = {GG, BB, GB, BG}

Problem 5 :

The order in which 4 blocks A, B, C and D can be lined up.

Solution :

Keeping A first,

ABCD, ABDC, ACBD, ACDB, ADBC, ADCB

Keeping B first 

BACD, BADC, BCDA, BCAD, BDAC, BDCA

Keeping C first 

CABD, CADB, CBDA, CBAD, CDAB, CDBA

Keeping D first 

DABC, DACB, DBAC, DBCA, DCAB, DCBA

Sample space

= {ABCD, ABDC, ACBD, ACDB, ADBC, ADCB, BACD, BADC, BCDA, BCAD, BDAC, BDCA, CABD, CADB, CBDA, CBAD, CDAB, CDBA, DABC, DACB, DBAC, DBCA, DCAB, DCBA}

Problem 6 :

The 8 different 3 child families.

Solution :

Let G b be the girl child and B be the boy child.

Sample space = {GGG, BBB, GGB, GBG, BGG, BBG, BGB, GBB}

Problem 7 :

Spinning a coin :

i) twice    ii)  three times     iii) four times

Solution : 

Let H be the head and T be the tail.

a) A coin is spinning twice

Sample space = {HH, TT, HT, TH}

b) A coin is spinning three times

Sample space = {HHH, TTT, HHT, HTH, THH, TTH, THT, HTT}

c) A coin is spinning four times

Sample space = {HHHH, TTTT, HHHT, HHTH, HTHH, THHH, HHTT, HTTH, TTHH, THTH, HTTT, THTT, TTHT, TTTH, HHHT, HTTT } 

Illustrate on a 2 dimensional grid the sample space for :

Problem 8 :

A rolling a die and tossing a coin simultaneously

Solution :

• When rolling a coin, we will get two outcomes. Head and tail.
• When rolling a die, we will get 6 outcomes, 1, 2, 3, 4, 5, 6 

By converting the list, 

{1H, 2H, 3H, 4H, 5H, 6H, 1T, 2T, 3T, 4T, 5T, 6T}

Problem 9 :

Rolling two dice

Solution :

By rolling each die, we will get 6 outcomes.

Outcomes are 1, 2, 3, 4, 5, 6

By converting it as list, 

{(1, 1)(1, 2)

Problem 10 :

Rolling a die and spinning a spinner with sides A, B, C, D

Solution :

• When we roll a die, the possible outcomes are 1, 2, 3, 4, 5, 6.
• When we roll a spinner, the possible outcomes are A, B, C and D.

Plotting the outcomes in horizontal and vertical axis.

By converting it as list, 

{(1, A) (1, B) (1, C) (1, D) (2, A) (2, B) (2, C) (2, D) (3, A) (3, B) (3, C) (3, D) (4, A) (4, B) (4, C) (4, D) (5, A) (5, B) (5, C) (5, D) (6, A) (6, B) (6, C) (6, D) }

Problem 11 :

Twirling two square spinners : 

One labelled A, B, C, D and the other 1, 2, 3, 4

Solution :

Outcomes of first spinner = A, B, C, D

Outcomes of second spinner = 1, 2, 3, 4

Sample space = {(A, 1) (A, 2) (A, 3) (A, 4) (B, 1) (B, 2) (B, 3) (B, 4) (C, 1) (C, 2) (C, 3) (C, 4) (D, 1) (D, 2) (D, 3) (D, 4)}

Illustrate on a tree diagram the sample space for

Problem 12 :

Tossing a 5 cent and 10 cent coin simultaneously.

Solution :

• A 5 cent coin will have two faces, head and tail
• A 10 cent coin will have two faces, head and tail

The possible outcomes are {HH, TT, HT, TH} 

Problem 13 :

Tossing a coin and twirling an equilateral triangular spinner labelled A, B and C.

Solution :

• When we toss a coin, we get two outcomes head and tail
• When we are twirling a spinner, the possible outcomes are A, B, C and D.

By converting it as list,

The possible outcomes are {(H, A) (H, B) (H, C) (T, A) (T, B) (T, C)   

Problem 14 :

Draw a venn diagram to show a class of 20 students where 7 study History and Geography, 10 study History, 15 study Geography and 2 study neither subject.

Solution :