1. 1. 90% 10.1 55, 67, 73, 63, 80, 49, 72, 63, 59, 46 46 80 46 80 (80 - 46) = 34 = ( ) () 41-50 3 51-60 7 61-70 14 71-80 8 41 80 (80 - 41) = 39
2. 2. 10.2 N x1, x2, x3 xN x = 2, 4, 6, 8, 10, 12, 14, 16 (2+ 4+ 6+ 8+ 10+ 12+ 14+ 16)/8 = 9 , = (|2-9|+|4-9|+|6-9|+|8-9|+|10-9|+|12-9|+|14-9|+|16-9|)/8 = (|-7|+|-5|+|-3|+|-1|+|1|+|3|+|5|+|7|)/8 = 4. ***[ 16-2=14, 4] x1, x2, x3 xN , f1, f2, f3 fN x =(1/) |xi x| 1 fi n= i 1 . 10 - 15 3 15 - 20 5 20 - 25 7 25 - 30 4 30 - 35 2
3. 3. xi fi xi fi |xi - x| |xi - x| fi 10 - 15 12.5 3 37.5 9.286 27.858 15 - 20 17.5 5 87.5 4.286 21.43 20 - 25 22.5 7 157.5 0.714 4.998 25 - 30 27.5 4 110 5.714 22.856 30 - 35 32.5 2 65 10.714 21.428 n= 21 1 xifi = 457.5 |xi x| 1 fi = 98.57 n= 21, x = 457.5/21 = 21.79 = 98.57/21321 = 4.69 10.3 n x1, x2, x3 xn s2 s 2, 4, 6, 8, 10, 12, 14, 16 )2 +4 +6 +8 +10 +12 +14 +16/(8 =9 , s2 = {(2-9)2 +(4-9)2 +(6-9)2 +(8-9)2 +(10-9)2 +(12-9)2 +
4. 4. (14-9)2 +(16-9)2 }/8 = (72 + 52 + 32 + 12 + 12 + 32 + 52 + 72 )/8 = 42 s = 42 = 6.48 x1, x2, x3 xN ,f1, f2, f3 fN x s2 = (1/) (xi x) 1 2 fi n= i 1 25-30 5 30-35 10 35-40 25 40-45 8 45-50 2 fi ( xi) xi*fi d= xi x d2 fi*d2 25-30 5 28 140 -9.2 84.64 423.2 30-35 10 33 330 -4.2 17.64 176.4 35-40 25 38 950 0.8 0.64 16 40-45 8 43 344 5.8 33.64 269.12 45-50 2 48 96 10.8 116.64 233.28 n = fi = 50 (xi*fi)=1860 (fi*d2 )=1118 = x = (xi*fi)/n = 1860/50 = 37.2 s2 = 1118/50 = 22.36 s= 22.36 = 4.728
5. 5. ( ): A X u=(X-A)/h , h s= h* [ (f*u2 )/n -( (f*u)/n2 ] f 25-30 5 30-35 12 35-40 20 40-45 8 45-50 5 (fi) (xi) ui=(xi-A)/i fiui fiui 2 25-30 5 28 -2 -10 20 30-35 12 33 -1 -12 12 35-40 20 38 0 0 0 40-45 8 43 1 8 8 45-50 5 48 2 10 20 n = fi = 50 fiui= -4 fiui 2 =60 s= h* [ (f*u2 )/n -( (f*u)/n2 ] = 5* [(60/50) - {-4/50}2 ] =5 * [1.1936] = 5.4625 = s2 = (5.4625)2 = 29.839
6. 6. 100 50 32 50 38 34 32 38 34 10.4 , 100 50/50 49/51 48/52 4/1 3/2 0.5 1.0 10.5
7. 7. 1, 2, 3, 4, 5, 6 A={1,2,3,4,5,6} , 1, 2, 3, 4, 5, 6 1 6/6 S={1,2,3,4,5,6} A ={2,4,6} Ac A= {1,3,5} 2,4,6
8. 8. 7 = / , 1, 2, 3, 4, 5, 6 2, 3, 5 (1- ) =3/6=0.5 1. A P(A) 1 0 P(A) 1 2. A P(A) = 0 3. 1 A P(A) = 1 4. A P(A) A P(A) P(A) + P(A) = 1 10.6 1: 10 (i) ? (ii) 10 8 ? (iii) 8 ?
9. 9. 210 (i) 1 1/210 (ii) 8 8 10 8 , 10 8 10 C8 22 (10 C8*22 ) (10 C8*22 )/210 (iii) 8 10 8 10 C8 1 10 C8/210 2: 4 6 (i) (ii) (iii) (iv) ? (i) 4/10 3 9, 3/9 4/10* 3/9= 2/15 (ii) (6/10)*(4/9)=4/15 (iii) (4/10)*(6/9)= 4/15 (iv) 6 4 6 C2*4 C1 10 2 10 C2 (6 C2*4 C1)/10 C2
10. 10. 5 10.7 A B , N = n1 + n2+ n3 + n4 A n(A), B n(B), n(AB), n(AB) , n(AB)=n(A)+n(B)-n(AB) , n(AB)/N = n(A)/N + n(B)/N -n(AB)/N , P(AB ) = P(A)+P(B)-P(AB) A, B, C P(ABC) =P(A)+P(B)+P(C)-P(AB)-P(BC)-P(CA)+P(ABC) 180 70 , 50 25 30 , 10 5 ? n(BEM ) = n(B) + n(E) + n(M )-n(BE)-n(BM )-n(EM )+n(BEM ) n(B) = 70; n(E) = 50; n(M ) = 25; n(BE) = 30; n(BM ) = 10; n(EM ) = 5; n(BEM ) = 0 n(BEM ) = 70 + 50 + 25 30 10 5 + 0 = 100 100 ABA B n1 n2 n3 n4
11. 11. (180 100) = 80 80/180 67% 80% ? A B P(A) P(B) P(A B) = P(A)+ P(B) , n(AB)/N = n(A)/N + n(B)/N - n(AB)/N n(AB) = 0, n(AB)/N = n(A)/N + n(B)/N , P(AB) = P(A) + P(B) , A1, A2, A3, P(A1 A2 A3 ) = P(A1 ) + P(A2 ) + P( A3) 52 ? 13 P(S) = 13/52= 13/52=1/4 P(H) = 13/52=1/4 P(HS) = 1/4+1/4= A+ 0.9 A+ 0.6 A+ ?
12. 12. A+ P(B)= 0.9, A+ P(E)= 0.6 A+ P(BE) = 0.9*0.6=0.54 A+ P(BE)= P(B)+ P(E) - P(BE)= 0.9+0.6- 0.54=0.96 10.8 A B P(A B) = P(AB) = P(A) * P(B) , A1, A2, A3... P(A1 A2 A3...) = P(A1)*P(A2)*P(A3)... P(AB) = P(A) * P(B/A) ; P(B/A) A B A B A , AB A B = (AB )/( A ) = ((AB )/N) / (( A )/N) [N = ] P(B/A) = P(AB) / P(A) , P(AB) = P(A) * P(B/A) 1: ? 1/6 1/2 =(1/6)*(1/2)= 1/12 2: 30% 5% ? P(C) = 0.05
13. 13. P(H/C) = 0.3 P(HC) = P(C)* P(H/C) = 0.05*0.3 = 0.015 10.9 1: , ? 1 /7 2 /7 1 /7*2 /7 = 2 /49 49 /2 25 2: 10% P(A)= 10%=0.1 P(GA)= 8000 /160000000 =0.00005 [ P(GA) ] P(G/A) = P(GA)/P(A)= 0.00005/0.1=0.0005 = 0. 05% 3: 30 45 ?
14. 14. P1= (1000-30)/1000=0.97 P2=(1000-45)/1000=0.955 = P1* P2 =0.97*0.955=0.92635 10.10 1. 14 17,17, 18, 20, 21, 23, 24, 25, 25, 28, 31, 33, 34 37 , , 2. , 20-25 12 26-30 14 31-35 33 36-40 25 41-45 21 46-50 31 51-55 26 56-60 20 3. 1 2 3 4 5 6 7 8 9 10 43 48 65 57 31 60 37 48 78 59 4.
15. 15. [0, 2) 4 [2, 4) 11 [4, 6) 24 [6, 8) 34 [8, 10) 40 5. 50000 4 45000 11 42000 24 40000 34 35000 40 6. 2, 3, 6, 8, 11,12, 6, 7, 3, 15, 10, 18, 5 10.11 1. , 2. 30 17 5 4 ? 3.
16. 16. 4. 6 5. 6. 5 ? 7. 3 8. (i) (ii) (iii) (iv) 9. ? 10. MISSISSIPPI (i) (ii) (iii) I, S, P 11. A B 0.3 0.5 0.65 , ,
17. 17. 12. ? 13. ? 14. ? 15. - - ? 1. (i) , ? (ii) ? (iii) 2, 4, 5 ? 2. (i) (ii) ? (iii) ? 3. (i) 2 ? (ii) , , , (iii) 4. (i) 0.6 , 0.7 0.4 (ii) ? (iii) ?
18. 18. 1. 1 20 3 5 ? (1) (2) 2 /5 (3) 8 /15 (4) 9 /20 2. , , ? (1) 10 /21 (2) 11 /21 (3) 2 /7 (4) 5 /7 3. ? (1) 3 /4 (2)1 /4 (3)3 /8 (4)7 /8 4. ? (1) 1(2) 1 /2 (3) 1 /4 (4) 3 /4 5. ? (1) 21 /46 (2) 25 /117 (3) 1 /50 (4) 3 /25 6. 2 ? (1) 1(2) 1 /12 (3) 1 /6 (4) 1 /2 7. 0.3 0.7 ? (1) 0.21(2) 1(3) 0.79 (4) 0.4 8. 0.3 0.7 ? (1) 0.21(2) 1(3) 0.79 (4) 0.4 9. 0.33 0.55 ? (1) 0.88(2) .22(3) 0.60 (4) 0.40 10. - (1) 1 /9 (2) 2 /9 (3) 4 /9 (4) 2 /9