Page 18 - Probability, Random Variables and Random Processes

PROBABILITY [CHAP 1

(b) The sample space S, contains 12 ordered pairs (i, j), i # j, 1 I i I 4, 1 I j I 4, where the first number
indicates the first number drawn. Thus,
(1, 2) (1, 3) (1, 4)
(2, 1) (2, 3) (2, 4)
(3, 1) (3, 2) (37 4)
(4, 1) (4, 2) (4, 3)

1.3. An experiment consists of rolling a die until a 6 is obtained.
(a) Find the sample space S, if we are interested in all possibilities.
(b) Find the sample space S, if we are interested in the number of throws needed to get a 6.
(a) The sample space S, would be

where the first line indicates that a 6 is obtained in one throw, the second line indicates that a 6 is
obtained in two throws, and so forth.
(b) In this case, the sample space S, is
S, = (i: i 2 1) = (1, 2, 3, ...)

where i is an integer representing the number of throws needed to get a 6.

1.4. Find the sample space for the experiment consisting of measurement of the voltage output v from
a transducer, the maximum and minimum of which are + 5 and - 5 volts, respectively.
A suitable sample space for this experiment would be

1.5. An experiment consists of tossing two dice.
(a) Find the sample space S.
(b) Find the event A that the sum of the dots on the dice equals 7.
(c) Find the event B that the sum of the dots on the dice is greater than 10.
(d) Find the event C that the sum of the dots on the dice is greater than 12.

(a) For this experiment, the sample space S consists of 36 points (Fig. 1-3):
S=((i,j):i,j=l,2,3,4,5,6)
where i represents the number of dots appearing on one die and j represents the number of dots
appearing on the other die.
(b) The event A consists of 6 points (see Fig. 1-3):
A = ((1, 6), (2, 51, (3, 4), (4, 31, (5, 2), (6, 1))
(c) The event B consists of 3 points (see Fig. 1-3):

(d) The event C is an impossible event, that is, C = 12(.