Tabular Representation of a Sample Space

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Year 10 Interactive Maths - Second Edition

Tabular Representation of a Sample Space

So, far, we have mainly considered simple probability experiments such as tossing a coin or throwing a die. Usually, probability experiments are more complicated. For example, tossing two different coins, tossing a coin and throwing a die, or tossing the same coin three times etc.

Recall the experiment of tossing a coin twice where we are interested in the number of heads. The tree diagram is shown below.

A tree diagram showing the possible outcomes and probabilities from tossing two coins.

Note that the tree diagram representation of this experiment involves two parts, 'the first toss of the coin' and 'the second toss of the coin'. Experiments that have two parts can be represented in tabular form.

For example, the following table uses rows to represent 'the first toss of the coin' and columns to represent 'the second toss of the coin'. The experiment's outcomes are shown in the bottom right-hand corner of the table where the rows and columns intersect.

The outcomes from the first toss of the coin are H and T and are shown vertically and the outcomes from the second toss of the coin are H and T and are shown horizontally. The resulting elements in the sample space for two coin tosses are HH, HT, TH, TT.

That is, S = {HH, HT, TH, TT}. Clearly, Pr(two heads) = 1/4, Pr(one head) = 2/4 = 1/2, Pr(no head) = 1/4

Example 7

Two dice are thrown. Let the events be defined as follows:

A = the numbers facing upwards on the two dice are the same
B = the sum of the numbers facing upwards on the two dice is 10
C = the sum of the numbers facing upwards on the two dice is 13

Find:

Solution:

Two dice are thrown. So, there are 36 elements in the sample space as shown in the table below.

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)}

(a) A = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}, Pr(A) = n(A)/n(S) = 6/36 = 1/6 (b) B = {(6,4), (5,5), (4,6)}, Pr(B) = n(B) / n(S) = 3/36 = 1/12 (c) A U B = {(1,1), (2,2), (3,3),(4,4), (5,5), (6,6), (6,4), (4,6)}, Pr(A U B) = n(A U B) / n(S) = 8/36 = 2/9 (d) A intersection B = {(5,5)}, Pr(A intersection B) = n(A intersection B) / n(S) = 1/36 (e) C = null set, Pr(C) = n(C) / n(S) = 0/36 = 0

Key Terms

tabular representation

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