21.12 Exercises 625
6.For the data in Exercise 1,(i) calculate and , (ii)use equations (21.7) and
(21.9) to compute the standard deviation and the skewness.
Section 21.4
7.A set of 10 balls consists of 6 red balls, 3 blue, and 1 yellow. If a ball is drawn at random,
find the probability that it is (i)red, (ii)yellow,(iii)red or yellow,(iv)not blue,(v)not
yellow.
8.A set of 50 numbered discs consists of 8 ones, 12 twos, 14 threes, 7 fours, and 9 fives.
If one disc is drawn at random, what is the probability that its number is (i)2, (ii)4,
(iii)2 or 4, (iv)≤ 1 4, (v)odd.
9.Find the probabilitiesP(2) toP(12) of all the possible outcomes of two throws of a die.
10.Find the probability of the following total scores from three throws of a die: (i)4, (ii)8,
(iii)4 or 8, (iv)more than 15.
11.A particle can be in three states with energiesε
0
,ε
1
, andε
2
(ε
0
1 < 1 ε
1
1 < 1 ε
2
), and probability
distribution at temperatureT. The quantity qis called the particle
partition function.
(i) Express qin terms of Tand the energies (use∑
iP
i1 = 11 ). (ii) Find the probability
distribution in the limit (a)T 1 → 10 , (b)T 1 → 1 ∞. (iii)Find the (combined) probability
distribution for a system of three independent particles.
Section 21.5
12.Find the probability that at least 3 heads are obtained from 5 tosses of (i)an unbiased
coin, (ii)a coin with probability 0.6 of coming up tails.
13.A system that can exist in a number of states with energiesE
0
1 < 1 E
1
1 < 1 E
2
1 <1-has
probability 0.1 of being in an excited state (withE 1 > 1 E
0
). Find the probability that in 10
independent observations, the system is found in the ground state (withE 1 = 1 E
0
) (i)every
time, (ii)only 5 times, (iii)at least 8 times.
14.Use the probability distribution of the outcomes of throwing a pair of dice, from
Exercise 9, to calculate the probability that two throws of a pair of dice have outcome
(i) 81 + 112 (one eight and one twelve), (ii) 91 + 111 , (iii) 101 + 110. Hence (iv)find the
probability of outcome 20 from two throws of two dice.
15.Use the probability distribution of the outcomes of throwing a pair of dice, from Exercise 9,
to calculate the probability that three throws of two dice have total outcome 30.
Section 21.6
16.List the permutations of 4 different objects.
17.List the permutations of 5 different objects taken 2 at a time.
18.List the combinations of 5 different objects taken 2 at a time.
19.List the distinct permutations of the 5 objects, A, A, A, B, and B.
20.What is the number of distinct permutations of 8 objects made up of 4 of type A, 3 of B,
and 1 of C?
21.Given an inexhaustible supply of objects A, B, and C, what is the number of distinct
permutations of these taken 8 at a time?
- (i) Given 3 distinguishable particles each of which can be in any of 4 states with
(different) energies E
1
, E
2
, E
3
, and E
4
, (a)what is the total number of ways of distributing
the particles amongst the states?, (b)how many states of the system have total energy
E
1
1 + 1 E
2
1 + 1 E
4
? (ii) Repeat (i)for 3 electrons instead of distinguishable particles.
Pe q
ikTi=
−
εx
3
xx,
2
,