## 1.2　Sample Space and Events

Suppose that we are about to perform an experiment whose outcome is not predictable in advance. However, while the outcome of the experiment will not be known in advance, let us suppose that the set of all possible outcomes is known. This set of all possible outcomes of an experiment is known as the sample space of the experiment and is denoted by S.

Some examples are the following.

1. If the experiment consists of the flipping of a coin, then

S = {H, T }

where H means that the outcome of the toss is a head and T that it is a tail. 2. If the experiment consists of rolling a die, then the sample space is

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

where the outcome i means that i appeared on the die, i = 1, 2, 3, 4, 5, 6.

1. If the experiments consists of flipping two coins, then the sample space consists of the following four points:

S = {(H, H ), (H, T ), (T, H ), (T, T )}

The outcome will be (H, H ) if both coins come up heads; it will be (H, T ) if the first coin comes up heads and the second comes up tails; it will be (T, H ) if the first comes up tails and the second heads; and it will be (T, T ) if both coins come up tails.

1. If the experiment consists of rolling two dice, then the sample space consists of the

following 36 points:

where the outcome (i, j ) is said to occur if i appears on the first die and j on the

second die.

1. If the experiment consists of measuring the lifetime of a car, then the sample space consists of all nonnegative real numbers. That is,

S = [0, ∞)∗ ■

Any subset E of the sample space S is known as an event. Some examples of events are the following.

1�. In Example (1), if E = {H }, then E is the event that a head appears on the flip of the coin. Similarly, if E = {T }, then E would be the event that a tail appears. 2�. In Example (2), if E = {1}, then E is the event that one appears on the roll of the die. If E = {2, 4, 6}, then E would be the event that an even number appears on the roll.

∗ The set (a, b) is defined to consist of all points x such that a < x < b. The set [a, b] is defined to consist of all points x such that a ≤ x ≤ b. The sets (a, b] and [a, b) are defined, respectively, to consist of all points x such that a < x ≤ b and all points x such that a ≤ x < b.

3�. In Example (3), if E = {(H, H ), (H, T )}, then E is the event that a head appears on the first coin.

4�. In Example (4), if E = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}, then E is the event that the sum of the dice equals seven.

5�. In Example (5), if E = (2, 6), then E is the event that the car lasts between two

and six years. ■

We say that the event E occurs when the outcome of the experiment lies in E . For any two events E and F of a sample space S we define the new event E ∪ F to consist of all outcomes that are either in E or in F or in both E and F . That is, the event E ∪ F will occur if either E or F occurs. For example, in (1) if E = {H } and F = {T }, then

E ∪ F = {H, T }

That is, E ∪ F would be the whole sample space S. In (2) if E = {1, 3, 5} and

F = {1, 2, 3}, then

E ∪ F = {1, 2, 3, 5}

and thus E ∪ F would occur if the outcome of the die is 1 or 2 or 3 or 5. The event E ∪ F is often referred to as the union of the event E and the event F . For any two events E and F , we may also define the new event EF, sometimes written E ∩ F , and referred to as the intersection of E and F , as follows. EF consists of all outcomes which are both in E and in F . That is, the event EF will occur only if both E and F occur. For example, in (2) if E = {1, 3, 5} and F = {1, 2, 3}, then

EF = {1, 3}

and thus EF would occur if the outcome of the die is either 1 or 3. In Example (1) if E = {H } and F = {T }, then the event EF would not consist of any outcomes and hence could not occur. To give such an event a name, we shall refer to it as the null event and denote it by Ø. (That is, Ø refers to the event consisting of no outcomes.) If EF = Ø, then E and F are said to be mutually exclusive.

We also define unions and intersections of more than two events⋃in a similar manner. If E1, E2 , . . . are events, then the union of these events, denoted by n=1 En , is defined to be the event that consists of all outcomes that are in En for at least ⋂ne value of n = 1, 2, . . . . Similarly, the intersection of the events En , denoted by ∞ n=1 E n , is defined to be the event consisting of those outcomes that are in all of the events En , n = 1, 2, . . .

Finally, for any event E we define the new event Ec , referred to as the complement of E , to consist of all outcomes in the sample space S that are not in E . That is, Ec will occur if and only if E does not occur. In Example (4) if E = {(1, 6), (2, 5), (3, 4), (4, 3), (5,2), (6, 1)}, then Ec will occur if the sum of the dice does not equal seven. Also note that since the experiment must result in some outcome, it follows that Sc = Ø.