152 PROBABILITY [CHAP. 7
BINOMIAL DISTRIBUTION
7.83.The probability that a women hits a target isp= 1 /3. She fires 50 times. Find the expected numberμof times she
will hit the target and the standard deviationσ.
7.84.TeamAhas probabilityp= 0 .8 of winning each time it plays. LetXdenote the number of timesAwill win in
n=100 games. Find the meanμ, varianceσ^2 , and standard deviationσofX.
7.85.An unprepared student takes a five-question true–false quiz and guesses every answer. Find the probability that the
student will pass the quiz if at least four correct answers is the passing grade.
7.86. LetXbe a binomially distributed random variableB(n, p)withE(X)=2 andVar(X)=^43. Findnandp.CHEBYSHEV’S INEQUALITY
7.87. LetXbe a random variable with meanμand standard deviationσ.
Use Chebyshev’s Inequality to estimateP(μ− 3 σ≤X≤μ+ 3 σ).
7.88. LetZbe the normal random variable with meanμ=0 and standard deviationσ=1.
Use Chebyshev’s Inequality to findavaluebfor whichP(−b≤Z≤b)= 0 .9.
7.89. LetXbe a random variable with meanμ=0 and standard deviationσ= 1 .5.
Use Chebyshev’s Inequality to estimateP(− 3 ≤X≤ 3 ).
7.90. LetXbe a random variable with meanμ=70.
For what value ofσwill Chebyshev’s Inequality giveP( 65 ≤X≤ 75 )≥ 0 .95.Answers to Supplementary Problems
The notation[x 1 ,...,xn;f(x 1 ),...,f(xn)]will be used
for the distributionf={(xi,f(Xi)}.
7.42. (a)A∩B∩CC; (c)(A∪B∪B)C=AC∩BCCC;
(b)(A∪C)∩BC; (d)(A∩B)∪(A∩C)∪(B∩C).
7.43. (a)n(S)=24;S={H,T}×{H,T}×{ 1 , 2 ,..., 6 }
(b)A={HH 2 ,HH 4 ,HH 6 };B={HH 2 ,HT 2 ,
TH2,TT 2 };C={HT 1 ,HT 3 ,HT 5 ,TH 1 ,TH 3 ,
TH 5 }
(c) (i)HH2; (ii)HT2,TH2,TT2; (iii).
7.44. (a) 3/6; (b) 15/16; (c) 20/36.
7.45. (a) 40/50; (b) 10/50; (c) 8/50; (d) 42/50.
7.46. (a) 1/15; (b) 7/15; (c) 8/15; (d) 7/15.
7.47. (a) 3/10; (b) 3/10; (c) 1/15; (d) 8/15.
7.48. 3 /5.
7.49. 1 /5.
7.50. (a) 3/4; (b) 1/4; (c) 1/3; (d) 7/12.
7.51. (c) and (d).
7.52. P(H)= 3 / 4 ;P(T)= 1 /4.
7.53. (a) 2/5; (b) 1/5; (c) 3/5.
7.54. (a) 0.6, 0.8, 0.5; (b) 0.5, 0.7, 0.4.
7.55. (a) 0.3; (b) 0.8; (c) 0.3; (d) 0.2.
7.56. (a) 1/6, 5/6; (b) 1/2, 1/3; (c) 1/2, 2/3; (d) 1/2,
(i) Yes; (ii) yes (iii) no.
7.57. (a) 12/30; (b) 4/30.
7.58. (a) 0.7; (b) 2/3; (c) 1/3.
7.59. (a) 1/3, 1/4; (b) yes.7.60. (a) 0.2; (b) 2/7; (c) 0.5.
7.61. (a)0.12,0.58; (b) 3/10, 4/10.
7.62. (a) 20%; (b) 1/3; (c) 1/2.
7.63. (a) 1/4; (b) 7/12.
7.64. (a) 3/4; (b) 1/3.
7.65. Only (A,B) are independent.
7.66. (a) l, (b) 0.
7.67. (a) 0.16; (b) 0.18.
7.68. (a) 6( 0. 3 )^2 ( 0. 7 )^2 = 0 .2646; (b) 1−( 0. 7 )^4 = 0 .7599.
7.69. (a) 10( 0. 4 )^2 ( 0. 6 )^3 ; (b) 1−( 0. 6 )^5.
7.70. (a) 1−( 2 / 3 )^5 = 211 /243; (b) Six times.
7.71. [1, 2, 3, 4, 5, 6; 11/36, 9/36, 7/36, 5/36, 3/36, 1/36];
E(X)= 91 / 36 ≈ 2 .5.
7.72. [0, 1, 2, 3, 4; 1/16, 7/16, 5/16, 2/16, 1/16];
E(X)= 27 / 16 ≈ 1 .7.
7.73. E= 1 .9.
7.74. (a) [0, 1, 2, 3; 1/64, 9/64, 27/64, 27/64];
(b)E(X)= 2 .25.
7.75. 23 / 8 ≈ 2 .9.
7.76. 11 / 9 ≈ 1 .2.
7.77. (a) 0.75; (b)− 0 .25.
7.78. 0.25.
7.79. (a)μ=4,σ^2 = 5 .5,σ= 2 .3; (b)μ=1,σ^2 = 2 .4,
σ= 1 .5.
7.80. μ=ap+bq;σ^2 =pq(a−b)^2 ;σ=|a−b|
√
pq