Biomedical Engineering Reference
In-Depth Information
choices for the first member, 9 choices for the second member, and 8 choices for the
third member. Thus, there are
N
(
N
- 1)(
N
-2)
720 ways of selecting
the three atoms for decay. However, these groups are not all distinct. In a counting
experiment, the decay of atoms 1, 3, and 8 in that order is not distinguished from
decay in the order 1, 8, and 3. The number of ways of ordering the three atoms that
decay is 3
=
10
×
9
×
8
=
×
2
×
1
=
3!
=
n
!
=
6. Therefore, the number of ways that any
n
=
3atoms
10 is given by the binomial coefficient
1)
can be chosen from among
N
=
N
n
10
3
10!
3!7!
=
10
×
9
×
8
720
6
= 120.
=
≡
=
(11.7)
3!
The probability that exactly 3 atoms decay is
10
3
p
3
q
7
P
3
=
=
120
×
0.00113
=
0.136.
(11.8)
(d) The probability that exactly 6 atoms decay in the 3 hours is
10
6
p
6
q
4
10!
6!4!
(0.154)
6
(0.846)
4
P
6
=
=
=
0.00143.
(11.9)
(e) The probability that no atom decays is
10
0
p
0
q
10
10!
0!10!
(0.846)
10
P
0
=
=
= 0.188.
(11.10)
Thus, there is a probability of 0.188 that no disintegrations occur in the 3-hour period,
which is approximately one-fourth of the half-life.
(f ) We can see from (c) and (d) that the general expression for the probability that
exactly
n
of the 10 atoms decay is
10
n
p
n
q
10-
n
.
P
n
=
(11.11)
(g) The sum of the probabilities for all possible numbers of disintegrations,
n
=
0
to
n
10, should be unity. From Eq. (11.11), and with the help of the last footnote for
the binomial expansion, we write
=
10
n
p
n
q
10-
n
10
10
(
p
+
q
)
10
.
P
n
=
=
(11.12)
n
=
0
n
=
0
Since
p
+
q
= 1, the total probability (11.12) is unity.
(h) With
N
=
100 atoms in the sample, the probability that none would decay in
the 3 h is
q
100
10
-8
. This is a much smaller probability than in
(e), where there are only 10 atoms in the initial sample. Seeing no atoms decay in a
sample of size 100 is a rare event.
(0.846)
100
=
=
5.46
×
1 The general expansion of a binomial to an
integral power
N
is given by
where
N
n
N
n
p
n
q
N
-
n
,
N
!
n
!(
N
-
n
)!
=
N
(
N
-1)
···
(
N
-
n
+1)
N
=
.
(
p
+
q
)
N
n
!
=
n
=
0
