Example 13.2.1

In a rolling a die trial (see example 13.1.1(1)) elementary events consists in receiving 1, 2, ... or 6 pips, Ω = {1, 2, 3, 4, 5, 6}.

1. Event A = "the number of received pips is even", is a subset of three elements {2, 4, 6} of the sample space Ω.
2. Event B = "more than 4 pips received", occurs if and only if we receive 5 or 6 pips, hence B = {5, 6}.
3. Event C = "not more that 4 pips received", occurs if and only if we receive 1, 2, 3 or 4 pips, hence C = {1, 2, 3, 4}.
4. Event D = "the number of pips received is a second power of a natural number" occurs if and only if we received 1 or 4 pips, hence D = {1, 4}.
5. Event E = "the number of pips received is congruent to 1 modulo 3", occurs if and only if the number of pips received devided by 3 has a remainder 1. Hence D = {1, 4}.

Let Ω be a sample space and A an event in this space, A ⊆ Ω. If A = {a₁, a₂, ..., a_n}, then elementary events a₁, a₂, ..., a_n are called events favorable to the event A.

We say that the event A is certain if A = Ω, and we say that the event is impossible if A = ∅. The certain event is absolutely guaranteed to happen every time, contrary to impossible event, that absolutely cannot happen.

Note. The sample space is not always finite and the events are not always the finite subsets of this space. In this lecture however, we will only consider a descrete case, all considered trials will have a finite sample space and all considered events will only have a finite number of the favorable events.

Fig. 13.2.1 The sample space in a trial of rolling two dice. The event A=" 6 received at least once", B= "the sum of pips is 7", and the intersection of the events A and D, where D= "5 received at least once" have been marked.

Example 13.2.2

The trial is rolling two distinguishable dice (marked for example as die number 1 and die number 2), we have the sample space containing of 36 elements Ω = {(i,j) : i, j = 1, 2, ...6} where, obviously, (i,j) is an elementary event indicating that i pips have been obtained after the first roll and j pips after the second, see Figure 13.2.1.

1. Event A = "6 received at least once", is a subset of the sample space composed of all the elementary events where there was 6 obtained after the first roll and all those elementary events where there was 6 after the second roll, hence A = {(6,i) : i = 1, 2, ...6} ∪ {(i,6) : i = 1, 2, ...5, 6}. This gives us 11 elementary events (6,1), (6,2), (6,3), (6,4), (6,5), (6,6), (1,6), (2,6), (3,6), (4,6), (5,6), favorable to the event A.
2. Event B = "the sum of pips is 7", is a subset {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}  of the sample space Ω. There are only 6 elementary events favorable to the event B.
3. Event C = "product of pips received is an even number", is a subset composed of all the elementary events excluding the following 9 events (1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3), (5,5).
4. Event "the sum of pips is not more than 12 " is a certain event as any elementary event has this property.
5. Event "the sum of pips is 1" is an impossible event as there is at least one pip at each die, which gives us at least 2 pips at both dice.

Question 13.2.1 How many elementary events are there favorable to the event "the number of pips on the first die is a divisor of the number of pips on the second die"?

« previous section

next section »