WRITING SAMPLE SPACE USING TREE DIAGRAMS EXAMPLES

Tree diagram allow us to see visually all possible outcomes of an random experiment. Each branch in a tree diagram represent a possible outcome.

(i) When we throw a die, then from the tree diagram the sample space can be written as

S = {1, 2, 3, 4, 5, 6 }

When we toss two coins, then fr om the tree diagram the sample space can be written as

S = {HH, HT, TH, TT}

Example 1 :

Write the sample space for tossing three coins using tree diagram.

Solution :

The required sample space

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

Total number of outcomes = 8.

Example 2 :

Write the sample space for selecting two balls from a bag containing 6 balls numbered 1 to 6 (using tree diagram).

Solution :

S = {(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)}

Total number of outcomes = 36

Example 3 :

If A is an event of a random experiment such that P(A) : P (A bar) = 17 : 15 and n(S) = 640 then find (i) P(A bar ) (ii) n(A).

Solution :

P(A) : P (A bar) = 17 : 15

P(A) / P (A bar) = 17 / 15

P(A) / [1 - P (A)] = 17 / 15

15 P(A) = 17 [1 - P(A)]

15 P(A) = 17 - 17 P(A)

15 P(A) + 17 P(A) = 17

32 P(A) = 17

P(A) = 17/32

P(A) = n(A) / n(S)

In order to make the denominator of p(A), let us multiply by 20

P(A) = (17/32) ⋅ (20/20)

P(A) = 340/640

P(A bar) = 1 - P(A)

= 1 - (340/640)

= 300/640

P(A bar) = 15/32

n(A) = 340

Example 4 :

A coin is tossed thrice. What is the probability of getting two consecutive tails?

Solution :

The required sample space

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

n(S) = 8

Let "A" be the event of getting two consecutive tails

A = { HTT, TTH, TTT }

n(A) = 3

P(A) = n(A) / n(S)

P(A) = 3/8

Example 5 :

Illustrate, using a tree diagram, the possible outcomes when:

a) drawing two marbles from a bag containing many red, green, and yellow marbles.

Solution :

Example 5 :

List the sample space for the following :

a) twirling a square spinner labelled A, B, C, D

b) the sexes of a 2-child family

c) the order in which 4 blocks A, B, C and D can be lined up

d) the 8 different 3-child families.

Solution :

a) twirling a square spinner labelled A, B, C, D

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

b) the sexes of a 2-child family

Let B be the boy child and G be girl child. Each family has 2 children. Then,

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

{c) A, B, C and D can be lined up

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

d) Let B be the boy child and G be girl child. Each family has 2 children. Then,

Sample space

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

Example 6 :

Illustrate on a 2-dimensional grid the sample space for :

a) rolling a die and tossing a coin simultaneously

b) rolling two dice

c) rolling a die and spinning a spinner with sides A, B, C, D

d) twirling two square spinners, one labelled A, B, C, D and the other 1, 2, 3, 4.

Solution :

a) Possible outcomes when we toss a coin = {H, T}

Possible outcomes when we throw die = {1,2, 3, 4, 5, 6}