Sample Space of an Event

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In the last article, we have studied the concept of probability. In this article, we shall study to write a sample space for a given event.

The Sample Space for Random Experiments:

From a group of 2 boys and 3 girls, two children are selected at random.

Two children out of the total 5 can be selected by ⁵C₂ ways = 10 ways
Let the two boys are B₁ and B₂ and the three girls are G₁, G_2, and G₃. The sample space is
S = { B₁B₂, B₁G₁, B₁G₂, B₁G₃, B₂G₁, B₂G₂, B₂G₃, G₁G₂, G₁G₃, G₂G₃}
n(S) = 10

A coin is tossed. If it shows head, we draw a ball from a bag consisting of 3 red and 4 black balls; if it shows a tail, we throw a die.

Let the three red balls are R₁, R_2, and R₃ and four black balls are B₁, B₂, B₃, and B₄. The sample space is
S = { (H, R₁), (H, R₂), (H, R₃), (H, B₁), (H, B₂), (H, B₃), (H, B₄), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)}
n(S) = 13

An experiment consists of rolling a die and tossing a coin once if the number on the die is even, If the number on the die is odd, the coin is tossed twice.

The sample space is

S = { (2, H), (2, T), (4, H), (4, T), (6, H), (6, T), (1, HH), (1, HT), (1, TH), (1, TT),

3, HH), (3, HT), (3, TH), (3, TT), (5, HH), (5, HT), (5, TH), (5, TT)}
n(S) = 18

A coin is tossed four times

The sample space is

S = { HHHH, HHHT, HTHH, THHH, HHTH, HHTT, HTTH, TTHH,

THHT, HTHT, THTH, TTTH, TTHT, THTT, HTTT, TTTT}
n(S) = 16

A coin is tossed and dice is thrown.

The sample space is

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

n(S) = 12

A coin is tossed and dice is rolled only in case the head is shown on the coin.

The sample space is

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

n(S) = 7

A coin is tossed twice and dice is rolled only in case the second throw is tail.

The sample space is

S = { HH, TH, (HT, 1), (HT, 2), (HT, 3), (HT, 4), (HT, 5), (HT, 6),

(TT, 1), (TT, 2), (TT, 3), (TT, 4), (TT, 5), (TT, 6)}

n(S) = 14

The experiment is of tossing a coin and tossing it for the second time if the head occurs. If the tail occurs on the first toss, then the die is tossed once.

The sample space is

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

n(S) = 8

A coin is tossed. If it shows a tail, we draw a ball from a box which contains 2 red and 3 black balls; if it shows head, we throw a die.

Let the two red balls are R₁ and R₂ and three balls are B1, B2 and B₃. The sample space is

S = { (T, R₁), (T, R₂), (T, B₁), (T, B₂), (T, B₃), (H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6)}

n(S) = 11

A coin is tossed repeatedly until a tail comes for the first time.

The sample space is

S = {T, HT, HHT, HHHT, HHHHT, ………..}

n(S) = ∞

A box contains 1 red and 3 black balls; Two balls are drawn at random in succession without replacement.

Let the one red ball be R₁ and three balls are B₁, B₂ and B₃. The sample space is

S = { (R₁, B₁), (R₁, B₂), (R₁, B₃), (B₁, R₁), (B₁, B₂), (B₁, B₃) , (B₂, B₁), (B₂, B₃), (B₃, B₁), (B₃, B₂),}

n(S) = 10

A pair of dice is rolled. If the outcome is a doublet, a coin is tossed. Determine the total number of elementary events associated with this experiment.

The sample space is
S = {(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2. 3), (2, 4),

(2, 5). (2, 6), (3, 1), (3, 2), (3, 4), (3, 5), (3, 6), (4, 1),

(4, 2), (4, 3), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4),

(5, 6), (6, 1I), (6, 2), (6, 3). (6, 4), (6, 5), (1, 1, H), (1, 1, T),

(2, 2, H), (2, 2, T), (3, 3, H), (3, 3, T), (4, 4, H), (4, 4, T),

(5, 5, H), (5, 5, T), (6, 6, H), (6, 6, T) }

n(S) = 42

A coin is tossed twice. If the second draw results in a head, a die is rolled.

The sample space is

S = { TT, HT, (HH, 1), (HH, 2), (HH, 3), (HH, 4), (HH, 5), (HH, 6), (TH, 1), (TH, 2), (TH, 3), (TH, 4), (TH, 5), (TH, 6),}

n(S) = 14

A bag contains 4 identical red balls and 3 identical black balls. The experiment consists of drawing a ball from the bag and replacing back in the box; then again drawing a ball.

The sample space is S = { RR, RB, BR, BB}

n(S) = 4

Note: As balls are identical no suffixes are used.

In random sampling, three items are selected from a lot. Each item is tested and classified as defective (D) or non-defective (N).

The sample space is

S = { DDD, DDN, DND, NDD, DNN, NDN, NND, TTT}

n(S) = 8

An experiment consists of boy girl composition of families with two children. We are interested in knowing whether it is a boy or a girl in the order of their birthdays.

The sample space is

S = { (B₁, B₂), (B₁, G₂), (G₁, B₂), (G₁, G₂)}

n(S) = 4

An experiment consists of a boy-girl composition of families with two children. We are interested in the number of boys in the family.

The sample space is

S = { 0, 1, 2}

n(S) = 3

There are three coloured dice of red, white and black colour. These dice are placed in a bag. One is drawn at random and rolled its colour and the number on the uppermost face is noted.

The sample space is

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

(W, 4), (W, 5), (W, 6), (B, 1), (B, 2), (B, 3), (B, 4), (B, 5), (B, 6)}

n(S) = 18

2 boys and 2 girls are in room P and 1 boy and three girls are in room Q. A person is selected from one of the rooms.

Let the two boys in the room P be B₁, B₂ and that in the room Q be B₃.

Let the two girls in the room P be G₁, G₂ and those in the room Q be G₃, G₄ and G₅.

The sample space is

S = { (P, B₁), (P, B₂), (P, G₁), (P, G₂), (Q, B₃), (Q, G₃), (Q, G₄), (Q, G₅)}

n(S) = 8

A bag contains one white and one red ball. A ball is drawn from the bag. If ball drawn is white is replaced back in the bag and again a ball is drawn from the bag; otherwise, a die is rolled.

The sample space is

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

n(S) = 8

A bag contains 1 white and 3 identical black balls. Two balls are drawn at random from the bag in succession.

The sample space is

S = { WB, BW, BB}

n(S) = 3

The numbers 1, 2, 3 and 4 are written separately on four slips of paper. The slips are then put in a box and mixed thoroughly. A person draws two slips from a box, one after the other, without replacement.

The sample space is

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

n(S) = 12

A coin is tossed. If the result is head, a die is thrown. If the die shows an even number, the die is thrown again.

The sample space is

S = { T, (H, 1), (H, 3), (H, 5), (H, 2, 1), (H, 2, 2), (H, 2, 3), (H, 2, 4),

(H, 2, 5), (H, 2, 6), (H,4, 1), (H, 4, 2), (H, 4, 3), (H, 4, 4),

(H, 4, 5), (H, 4, 6), (H, 6, 1), (H, 6, 2), (H, 6, 3),

(H, 6, 4), (H, 6, 5), (H, 6, 6),}

n(S) = 22

In the next article, we shall study the meaning of events and their types and also to write event space for the given experiment.

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