Summaries
14th February
2011
Session 1.7
Marginal, Joint and
Conditional Probabilities
Suppose that we have
fair dice: d4 with face values {1,2,3,4}, d6 with face values {1,2,3,4,5,6} and
d8 with face values {1,2,3,4,5,6,7,8}. Our experiment consists of first
randomly selecting one of the dice and then tossing that die and noting the
face value.
The first stage
probabilities:
Pr{Select d4} = 1/3( = P4)
Pr{Select d6} = 1/3( = P6)
Pr{Select d8} = 1/3( = P8)
The conditional probabilities:
Pr{1 shows | d4 selected} = 1/4
Pr{2 shows | d4 selected} = 1/4
Pr{3 shows | d4 selected} = 1/4
Pr{4 shows | d4 selected} = 1/4
Pr{1 shows | d6 selected} = 1/6
Pr{2 shows | d6 selected} = 1/6
Pr{3 shows | d6 selected} = 1/6
Pr{4 shows | d6 selected} = 1/6
Pr{5 shows | d6 selected} = 1/6
Pr{6 shows | d6 selected} = 1/6
Pr{1 shows | d8 selected} = 1/8
Pr{2 shows | d8 selected} = 1/8
Pr{3 shows | d8 selected} = 1/8
Pr{4 shows | d8 selected} = 1/8
Pr{5 shows | d8 selected} = 1/8
Pr{6 shows | d8 selected} = 1/8
Pr{7 shows | d8 selected} = 1/8
Pr{8 shows | d8 selected} = 1/8
The joint
probabilities
Pr{1 shows} = Pr{1 shows | d4 selected}*Pr{d4 selected} + Pr{1 shows | d6 selected}*Pr{d6 selected} + Pr{1 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/4) + (1/3)*(1/6) + (1/3)*(1/8) = (1/3)*(13/24) = 13/72 ≈ 0.1806
Pr{2 shows} = Pr{2 shows | d4 selected}*Pr{d4 selected} +
Pr{2 shows | d6 selected}*Pr{d6 selected} + Pr{2 shows | d8 selected}*Pr{d8 selected}
= (1/3)*(1/4) + (1/3)*(1/6) + (1/3)*(1/8) = (1/3)*(13/24) = 13/72 ≈ 0.1806
Pr{3 shows} = Pr{3 shows | d4 selected}*Pr{d4 selected} +
Pr{3 shows | d6 selected}*Pr{d6 selected} + Pr{3 shows | d8 selected}*Pr{d8
selected} = (1/3)*(1/4) + (1/3)*(1/6) + (1/3)*(1/8) = (1/3)*(13/24) = 13/72 ≈
0.1806
Pr{4 shows} = Pr{4 shows | d4 selected}*Pr{d4 selected} +
Pr{4 shows | d6 selected}*Pr{d6 selected} + Pr{4 shows | d8 selected}*Pr{d8
selected} = (1/3)*(1/4) + (1/3)*(1/6) + (1/3)*(1/8) = (1/3)*(13/24) = 13/72 ≈
0.1806
Pr{5 shows} = Pr{5 shows | d6 selected}*Pr{d6 selected} + Pr{5 shows | d8
selected}*Pr{d8 selected} = (1/3)*(1/6) + (1/3)*(1/8) = (1/3)*(7/24) = 7/72 ≈
0.0972
Pr{6 shows} = Pr{6 shows | d6 selected}*Pr{d6 selected} + Pr{6 shows | d8
selected}*Pr{d8 selected} = (1/3)*(1/6) + (1/3)*(1/8) = (1/3)*(7/24) = 7/72 ≈
0.0972
Pr{7 shows} = Pr{7 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/8) = 3/72 ≈
0.0417
Pr{8 shows} = Pr{8 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/8) = 3/72 ≈
0.0417
Sample Tables from Fall 2010
Compare sample
proportions (p) to probabilities (P).
6:30
|
Sample 1 |
D8 |
D6 |
D4 |
Joint |
|||||||
|
Face Value |
n |
p |
Face Value |
n |
p |
Face Value |
n |
p |
n |
p |
P |
|
1 |
7 |
0.104477612 |
1 |
13 |
0.2363636 |
1 |
23 |
0.2948718 |
43 |
0.215 |
0.1806 |
|
2 |
10 |
0.149253731 |
2 |
13 |
0.2363636 |
2 |
17 |
0.2179487 |
40 |
0.2 |
0.1806 |
|
3 |
9 |
0.134328358 |
3 |
8 |
0.1454545 |
3 |
15 |
0.1923077 |
32 |
0.16 |
0.1806 |
|
4 |
5 |
0.074626866 |
4 |
6 |
0.1090909 |
4 |
23 |
0.2948718 |
34 |
0.17 |
0.1806 |
|
5 |
4 |
0.059701493 |
5 |
12 |
0.2181818 |
|
|
16 |
0.08 |
0.0972 |
|
|
6 |
11 |
0.164179104 |
6 |
3 |
0.0545455 |
|
|
14 |
0.07 |
0.0972 |
|
|
7 |
13 |
0.194029851 |
|
|
|
|
|
13 |
0.065 |
0.0417 |
|
|
8 |
8 |
0.119402985 |
|
|
|
|
|
8 |
0.04 |
0.0417 |
|
|
Total |
67 |
1 |
Total |
55 |
1 |
Total |
78 |
1 |
200 |
1 |
1 |
|
Sample 2 |
D8 |
D6 |
D4 |
Joint |
|||||||
|
Face Value |
n |
p |
Face Value |
n |
p |
Face Value |
n |
p |
n |
p |
P |
|
1 |
7 |
0.112903226 |
1 |
6 |
0.0895522 |
1 |
9 |
0.1267606 |
22 |
0.11 |
0.1806 |
|
2 |
8 |
0.129032258 |
2 |
8 |
0.119403 |
2 |
24 |
0.3380282 |
40 |
0.2 |
0.1806 |
|
3 |
5 |
0.080645161 |
3 |
19 |
0.2835821 |
3 |
18 |
0.2535211 |
42 |
0.21 |
0.1806 |
|
4 |
6 |
0.096774194 |
4 |
12 |
0.1791045 |
4 |
20 |
0.2816901 |
38 |
0.19 |
0.1806 |
|
5 |
6 |
0.096774194 |
5 |
11 |
0.1641791 |
|
|
17 |
0.085 |
0.0972 |
|
|
6 |
11 |
0.177419355 |
6 |
11 |
0.1641791 |
|
|
22 |
0.11 |
0.0972 |
|
|
7 |
8 |
0.129032258 |
|
|
|
|
|
8 |
0.04 |
0.0417 |
|
|
8 |
11 |
0.177419355 |
|
|
|
|
|
11 |
0.055 |
0.0417 |
|
|
Total |
62 |
1 |
Total |
67 |
1 |
Total |
71 |
1 |
200 |
1 |
1 |
|
Sample 3 |
D8 |
D6 |
D4 |
Joint |
|||||||
|
Face Value |
n |
p |
Face Value |
n |
p |
Face Value |
n |
p |
n |
p |
P |
|
1 |
4 |
0.06779661 |
1 |
14 |
0.1891892 |
1 |
17 |
0.2833333 |
35 |
0.1813472 |
0.1806 |
|
2 |
3 |
0.050847458 |
2 |
12 |
0.1621622 |
2 |
21 |
0.35 |
36 |
0.1865285 |
0.1806 |
|
3 |
7 |
0.118644068 |
3 |
11 |
0.1486486 |
3 |
8 |
0.1333333 |
26 |
0.134715 |
0.1806 |
|
4 |
7 |
0.118644068 |
4 |
13 |
0.1756757 |
4 |
14 |
0.2333333 |
34 |
0.1761658 |
0.1806 |
|
5 |
7 |
0.118644068 |
5 |
11 |
0.1486486 |
|
|
18 |
0.0932642 |
0.0972 |
|
|
6 |
7 |
0.118644068 |
6 |
13 |
0.1756757 |
|
|
20 |
0.1036269 |
0.0972 |
|
|
7 |
13 |
0.220338983 |
|
|
|
|
|
13 |
0.0673575 |
0.0417 |
|
|
8 |
11 |
0.186440678 |
|
|
|
|
|
11 |
0.0569948 |
0.0417 |
|
|
Total |
59 |
1 |
Total |
74 |
1 |
Total |
60 |
1 |
193 |
1 |
1 |
|
Sample 4 |
D8 |
D6 |
D4 |
Joint |
|||||||
|
Face Value |
n |
p |
Face Value |
n |
p |
Face Value |
n |
p |
n |
p |
P |
|
1 |
9 |
0.140625 |
1 |
9 |
0.1384615 |
1 |
15 |
0.2112676 |
33 |
0.165 |
0.1806 |
|
2 |
6 |
0.09375 |
2 |
12 |
0.1846154 |
2 |
21 |
0.2957746 |
39 |
0.195 |
0.1806 |
|
3 |
7 |
0.109375 |
3 |
7 |
0.1076923 |
3 |
24 |
0.3380282 |
38 |
0.19 |
0.1806 |
|
4 |
6 |
0.09375 |
4 |
14 |
0.2153846 |
4 |
11 |
0.1549296 |
31 |
0.155 |
0.1806 |
|
5 |
7 |
0.109375 |
5 |
13 |
0.2 |
|
|
20 |
0.1 |
0.0972 |
|
|
6 |
10 |
0.15625 |
6 |
10 |
0.1538462 |
|
|
20 |
0.1 |
0.0972 |
|
|
7 |
10 |
0.15625 |
|
|
|
|
|
10 |
0.05 |
0.0417 |
|
|
8 |
9 |
0.140625 |
|
|
|
|
|
9 |
0.045 |
0.0417 |
|
|
Total |
64 |
1 |
Total |
65 |
1 |
Total |
71 |
1 |
200 |
1 |
1 |
|
Sample 5 |
D8 |
D6 |
D4 |
Joint |
|||||||
|
Face Value |
n |
p |
Face Value |
n |
p |
Face Value |
n |
p |
n |
p |
P |
|
1 |
5 |
0.080645161 |
1 |
8 |
0.1269841 |
1 |
22 |
0.2933333 |
35 |
0.175 |
0.1806 |
|
2 |
9 |
0.14516129 |
2 |
9 |
0.1428571 |
2 |
17 |
0.2266667 |
35 |
0.175 |
0.1806 |
|
3 |
13 |
0.209677419 |
3 |
13 |
0.2063492 |
3 |
10 |
0.1333333 |
36 |
0.18 |
0.1806 |
|
4 |
10 |
0.161290323 |
4 |
9 |
0.1428571 |
4 |
26 |
0.3466667 |
45 |
0.225 |
0.1806 |
|
5 |
7 |
0.112903226 |
5 |
11 |
0.1746032 |
|
|
18 |
0.09 |
0.0972 |
|
|
6 |
2 |
0.032258065 |
6 |
13 |
0.2063492 |
|
|
15 |
0.075 |
0.0972 |
|
|
7 |
9 |
0.14516129 |
|
|
|
|
|
9 |
0.045 |
0.0417 |
|
|
8 |
7 |
0.112903226 |
|
|
|
|
|
7 |
0.035 |
0.0417 |
|
|
Total |
62 |
1 |
Total |
63 |
1 |
Total |
75 |
1 |
200 |
1 |
1 |
|
Sample 6 |
D8 |
D6 |
D4 |
Joint |
|||||||
|
Face Value |
n |
p |
Face Value |
n |
p |
Face Value |
n |
p |
n |
p |
P |
|
1 |
3 |
0.046875 |
1 |
14 |
0.2413793 |
1 |
21 |
0.2692308 |
38 |
0.19 |
0.1806 |
|
2 |
10 |
0.15625 |
2 |
13 |
0.2241379 |
2 |
18 |
0.2307692 |
41 |
0.205 |
0.1806 |
|
3 |
14 |
0.21875 |
3 |
9 |
0.1551724 |
3 |
21 |
0.2692308 |
44 |
0.22 |
0.1806 |
|
4 |
14 |
0.21875 |
4 |
6 |
0.1034483 |
4 |
18 |
0.2307692 |
38 |
0.19 |
0.1806 |
|
5 |
9 |
0.140625 |
5 |
7 |
0.1206897 |
|
|
16 |
0.08 |
0.0972 |
|
|
6 |
7 |
0.109375 |
6 |
9 |
0.1551724 |
|
|
16 |
0.08 |
0.0972 |
|
|
7 |
3 |
0.053571429 |
|
|
|
|
|
3 |
0.015 |
0.0417 |
|
|
8 |
4 |
0.0625 |
|
|
|
|
|
4 |
0.02 |
0.0417 |
|
|
Total |
64 |
1.006696429 |
Total |
58 |
1 |
Total |
78 |
1 |
200 |
1 |
1 |
|
Pooled |
D8 |
D6 |
D4 |
Joint |
|||||||
|
Face Value |
n |
p |
Face Value |
n |
p |
Face Value |
n |
p |
n |
p |
P |
|
1 |
35 |
0.092592593 |
1 |
64 |
0.1675393 |
1 |
107 |
0.2471132 |
206 |
0.1726739 |
0.1806 |
|
2 |
46 |
0.121693122 |
2 |
67 |
0.1753927 |
2 |
118 |
0.2725173 |
231 |
0.1936295 |
0.1806 |
|
3 |
55 |
0.145502646 |
3 |
67 |
0.1753927 |
3 |
96 |
0.221709 |
218 |
0.1827326 |
0.1806 |
|
4 |
48 |
0.126984127 |
4 |
60 |
0.1570681 |
4 |
112 |
0.2586605 |
220 |
0.1844091 |
0.1806 |
|
5 |
40 |
0.105820106 |
5 |
65 |
0.1701571 |
|
|
105 |
0.0880134 |
0.0972 |
|
|
6 |
48 |
0.126984127 |
6 |
59 |
0.1544503 |
|
|
107 |
0.0896899 |
0.0972 |
|
|
7 |
56 |
0.148148148 |
|
|
|
|
|
56 |
0.0469405 |
0.0417 |
|
|
8 |
50 |
0.132275132 |
|
|
|
|
|
50 |
0.0419111 |
0.0417 |
|
|
Total |
378 |
1 |
Total |
382 |
1 |
Total |
433 |
1 |
1193 |
1 |
1 |
8:00
|
Sample 1 |
D8 |
D6 |
D4 |
Joint |
|||||||
|
Face Value |
n |
p |
Face Value |
n |
p |
Face Value |
n |
p |
n |
p |
P |
|
1 |
6 |
0.08108108 |
1 |
12 |
0.20689655 |
1 |
19 |
0.26760563 |
37 |
0.18226601 |
0.1806 |
|
2 |
9 |
0.12162162 |
2 |
8 |
0.13793103 |
2 |
19 |
0.26760563 |
36 |
0.1773399 |
0.1806 |
|
3 |
11 |
0.14864865 |
3 |
8 |
0.13793103 |
3 |
15 |
0.21126761 |
34 |
0.16748768 |
0.1806 |
|
4 |
10 |
0.13513514 |
4 |
14 |
0.24137931 |
4 |
18 |
0.25352113 |
42 |
0.20689655 |
0.1806 |
|
5 |
12 |
0.16216216 |
5 |
7 |
0.12068966 |
|
|
19 |
0.09359606 |
0.0972 |
|
|
6 |
10 |
0.13513514 |
6 |
9 |
0.15517241 |
|
|
19 |
0.09359606 |
0.0972 |
|
|
7 |
11 |
0.14864865 |
|
|
|
|
|
11 |
0.05418719 |
0.0417 |
|
|
8 |
5 |
0.06756757 |
|
|
|
|
|
5 |
0.02463054 |
0.0417 |
|
|
Total |
74 |
1 |
Total |
58 |
1 |
Total |
71 |
1 |
203 |
1 |
1 |
|
Sample 2 |
D8 |
D6 |
D4 |
Joint |
|||||||
|
Face Value |
n |
p |
Face Value |
n |
p |
Face Value |
n |
p |
n |
p |
P |
|
1 |
5 |
0.06944444 |
1 |
6 |
0.10169492 |
1 |
21 |
0.30434783 |
32 |
0.16 |
0.1806 |
|
2 |
15 |
0.20833333 |
2 |
9 |
0.15254237 |
2 |
16 |
0.23188406 |
40 |
0.2 |
0.1806 |
|
3 |
4 |
0.05555556 |
3 |
10 |
0.16949153 |
3 |
15 |
0.2173913 |
29 |
0.145 |
0.1806 |
|
4 |
5 |
0.06944444 |
4 |
8 |
0.13559322 |
4 |
17 |
0.24637681 |
30 |
0.15 |
0.1806 |
|
5 |
15 |
0.20833333 |
5 |
11 |
0.18644068 |
|
|
26 |
0.13 |
0.0972 |
|
|
6 |
8 |
0.11111111 |
6 |
15 |
0.25423729 |
|
|
23 |
0.115 |
0.0972 |
|
|
7 |
13 |
0.18055556 |
|
|
|
|
|
13 |
0.065 |
0.0417 |
|
|
8 |
7 |
0.09722222 |
|
|
|
|
|
7 |
0.035 |
0.0417 |
|
|
Total |
72 |
1 |
Total |
59 |
1 |
Total |
69 |
1 |
200 |
1 |
1 |
|
Sample 3 |
D8 |
D6 |
D4 |
Joint |
|||||||
|
Face Value |
n |
p |
Face Value |
n |
p |
Face Value |
n |
p |
n |
p |
P |
|
1 |
11 |
0.10784314 |
1 |
15 |
0.25862069 |
1 |
9 |
0.225 |
35 |
0.175 |
0.1806 |
|
2 |
15 |
0.14705882 |
2 |
10 |
0.17241379 |
2 |
12 |
0.3 |
37 |
0.185 |
0.1806 |
|
3 |
22 |
0.21568627 |
3 |
8 |
0.13793103 |
3 |
8 |
0.2 |
38 |
0.19 |
0.1806 |
|
4 |
15 |
0.14705882 |
4 |
10 |
0.17241379 |
4 |
11 |
0.275 |
36 |
0.18 |
0.1806 |
|
5 |
10 |
0.09803922 |
5 |
4 |
0.06896552 |
|
|
14 |
0.07 |
0.0972 |
|
|
6 |
10 |
0.09803922 |
6 |
11 |
0.18965517 |
|
|
21 |
0.105 |
0.0972 |
|
|
7 |
12 |
0.11764706 |
|
|
|
|
|
12 |
0.06 |
0.0417 |
|
|
8 |
7 |
0.06862745 |
|
|
|
|
|
7 |
0.035 |
0.0417 |
|
|
Total |
102 |
1 |
Total |
58 |
1 |
Total |
40 |
1 |
200 |
1 |
1 |
|
Sample 4 |
D8 |
D6 |
D4 |
Joint |
|||||||
|
Face Value |
n |
p |
Face Value |
n |
p |
Face Value |
n |
p |
n |
p |
P |
|
1 |
7 |
0.12068966 |
1 |
11 |
0.14864865 |
1 |
18 |
0.26086957 |
36 |
0.17910448 |
0.1806 |
|
2 |
11 |
0.18965517 |
2 |
12 |
0.16216216 |
2 |
14 |
0.20289855 |
37 |
0.1840796 |
0.1806 |
|
3 |
6 |
0.10344828 |
3 |
17 |
0.22972973 |
3 |
18 |
0.26086957 |
41 |
0.2039801 |
0.1806 |
|
4 |
6 |
0.10344828 |
4 |
13 |
0.17567568 |
4 |
19 |
0.27536232 |
38 |
0.18905473 |
0.1806 |
|
5 |
10 |
0.17241379 |
5 |
9 |
0.12162162 |
|
|
19 |
0.09452736 |
0.0972 |
|
|
6 |
10 |
0.17241379 |
6 |
12 |
0.16216216 |
|
|
22 |
0.10945274 |
0.0972 |
|
|
7 |
4 |
0.06896552 |
|
|
|
|
|
4 |
0.0199005 |
0.0417 |
|
|
8 |
4 |
0.06896552 |
|
|
|
|
|
4 |
0.0199005 |
0.0417 |
|
|
Total |
58 |
1 |
Total |
74 |
1 |
Total |
69 |
1 |
201 |
1 |
1 |
|
Sample 5 |
D8 |
D6 |
D4 |
Joint |
|||||||
|
Face Value |
n |
p |
Face Value |
n |
p |
Face Value |
n |
p |
n |
p |
P |
|
1 |
7 |
0.11111111 |
1 |
7 |
0.10769231 |
1 |
20 |
0.26666667 |
34 |
0.16748768 |
0.1806 |
|
2 |
5 |
0.07936508 |
2 |
10 |
0.15384615 |
2 |
25 |
0.33333333 |
40 |
0.19704433 |
0.1806 |
|
3 |
13 |
0.20634921 |
3 |
13 |
0.2 |
3 |
13 |
0.17333333 |
39 |
0.19211823 |
0.1806 |
|
4 |
7 |
0.11111111 |
4 |
6 |
0.09230769 |
4 |
17 |
0.22666667 |
30 |
0.14778325 |
0.1806 |
|
5 |
10 |
0.15873016 |
5 |
19 |
0.29230769 |
|
|
29 |
0.14285714 |
0.0972 |
|
|
6 |
9 |
0.14285714 |
6 |
10 |
0.15384615 |
|
|
19 |
0.09359606 |
0.0972 |
|
|
7 |
5 |
0.07936508 |
|
|
|
|
|
5 |
0.02463054 |
0.0417 |
|
|
8 |
7 |
0.11111111 |
|
|
|
|
|
7 |
0.03448276 |
0.0417 |
|
|
Total |
63 |
1 |
Total |
65 |
1 |
Total |
75 |
1 |
203 |
1 |
1 |
|
Sample 6 |
D8 |
D6 |
D4 |
Joint |
|||||||
|
Face Value |
n |
p |
Face Value |
n |
p |
Face Value |
n |
p |
n |
p |
P |
|
1 |
12 |
0.17910448 |
1 |
9 |
0.13636364 |
1 |
19 |
0.27536232 |
40 |
0.1980198 |
0.1806 |
|
2 |
10 |
0.14925373 |
2 |
7 |
0.10606061 |
2 |
20 |
0.28985507 |
37 |
0.18316832 |
0.1806 |
|
3 |
10 |
0.14925373 |
3 |
14 |
0.21212121 |
3 |
17 |
0.24637681 |
41 |
0.2029703 |
0.1806 |
|
4 |
4 |
0.05970149 |
4 |
10 |
0.15151515 |
4 |
13 |
0.1884058 |
27 |
0.13366337 |
0.1806 |
|
5 |
4 |
0.05970149 |
5 |
10 |
0.15151515 |
|
|
14 |
0.06930693 |
0.0972 |
|
|
6 |
4 |
0.05970149 |
6 |
16 |
0.24242424 |
|
|
20 |
0.0990099 |
0.0972 |
|
|
7 |
13 |
0.22413793 |
|
|
|
|
|
13 |
0.06435644 |
0.0417 |
|
|
8 |
10 |
0.14925373 |
|
|
|
|
|
10 |
0.04950495 |
0.0417 |
|
|
Total |
67 |
1.03010808 |
Total |
66 |
1 |
Total |
69 |
1 |
202 |
1 |
1 |
|
Pooled |
D8 |
D6 |
D4 |
Joint |
|||||||
|
Face Value |
n |
p |
Face Value |
n |
p |
Face Value |
n |
p |
n |
p |
P |
|
1 |
48 |
0.11009174 |
1 |
60 |
0.15789474 |
1 |
106 |
0.2697201 |
214 |
0.17700579 |
0.1806 |
|
2 |
65 |
0.14908257 |
2 |
56 |
0.14736842 |
2 |
106 |
0.2697201 |
227 |
0.18775848 |
0.1806 |
|
3 |
66 |
0.15137615 |
3 |
70 |
0.18421053 |
3 |
86 |
0.21882952 |
222 |
0.18362283 |
0.1806 |
|
4 |
47 |
0.10779817 |
4 |
61 |
0.16052632 |
4 |
95 |
0.24173028 |
203 |
0.16790736 |
0.1806 |
|
5 |
61 |
0.13990826 |
5 |
60 |
0.15789474 |
|
|
121 |
0.10008271 |
0.0972 |
|
|
6 |
51 |
0.11697248 |
6 |
73 |
0.19210526 |
|
|
124 |
0.1025641 |
0.0972 |
|
|
7 |
58 |
0.13302752 |
|
|
|
|
|
58 |
0.04797353 |
0.0417 |
|
|
8 |
40 |
0.09174312 |
|
|
|
|
|
40 |
0.03308519 |
0.0417 |
|
|
Total |
436 |
1 |
Total |
380 |
1 |
Total |
393 |
1 |
1209 |
1 |
1 |
Conditional Probability
Conditional = Joint / Prior
Pr{A|B} = Pr{A∩B} / Pr{B}
How much of B is tied up in A ?
Case Study 1.11
Conditional Probability
Case Study Description: Compute
conditional probabilities associated with the color sequence experiment.
Suppose that we have a special box -
each time we press a button on the box, it prints out a sequence of colors, in
order - it prints four colors at a time. Suppose the box follows the following
Probabilities for each Color Sequence:
|
Color Sequence |
Probability CS Prints Out |
|
BBBB |
.10 = 10% |
|
BGGB |
.25 = 25% |
|
RGGR |
.05 = 05% |
|
YYYY |
.30 = 30% |
|
BYRG |
.15 = 15% |
|
RYYB |
.15 = 15% |
|
Total |
1.00 = 100% |
Let's define the experiment: We push
the button, and then the box prints out exactly one (1) of the above listed color
sequences. We then note the resulting (printed out) color sequence.
Compute Pr{
blue shows 1st | blue shows 4th };
Compute Joint
Probability
Pr{ B 1st and B 4th } = Pr{
exactly one of BBBB, BGGB shows } = Pr{ BBBB} + Pr{BGGB} =.10 + .25 = .35
Compute Prior Probability
Pr{ B 4th } = Pr{ exactly one of BBBB,
BGGB, RYYB shows } = Pr{BBBB} + Pr{BGGB} +
Pr{RYYB} = .10+.25+.15 = .50
Conditional Probability =
Joint Probability / Prior Probability
So, Pr{
B 1st | B 4th } = .35/ .50=
.70
Compute Pr{
green shows 2nd or 3rd | yellow shows };
Compute Joint
Probability
Pr{ G 2nd or 3rd and Y
shows } = 0, since no sequences meet this requirement
Compute Prior Probability
Pr{ Y shows } = Pr{ exactly
one of YYYY, BYRG, RYYB shows } = Pr{YYYY}+ Pr{BYRG}+ Pr{RYYB} = .30+.15+.15 =
.60
Conditional Probability =
Joint Probability / Prior Probability
So, Pr{
G 2nd or 3rd | Y shows } = 0 / .60= 0
Compute Pr{
yellow shows | red shows }.
Compute Joint
Probability
Pr{
Y and R show } = Pr{ exactly one of BYRG, RYYB shows } = Pr{BYRG}+
Pr{RYYB } = .15 + .15 = .30
Compute Prior Probability
Pr{ R
shows } = Pr{ exactly one of RGGR, BYRG, RYYB shows } = Pr{RGGR}+ Pr{BYRG}+
Pr{RYYB } = .05+.15+.15 = .35
Conditional Probability =
Joint Probability / Prior Probability
So, Pr{
Y shows | R shows } = .30/.35 = 6/7 = .8571
Case Study 1.12
Conditional Probability
II: Pair of Dice
Case Description: Compute conditional
probabilities.
Suppose we have a pair of fair dice:
d4(faces 1,2,3,4), d6(faces 1,2,3,4,5,6). In our experiment, we toss this pair
of dice, and note the face value from each die. For simplicity, we write the
outcome as (d4 result, d6 result). Assume that the dice operate independently
and separately.
Case Objectives:
Identify the simple (basic) events. Compute (and justify) a probability for each simple event.
As before, Pr{ ( any d4 face,
any d6 face) } = Pr{ any d4 face }*Pr{ any d6 face } = (1/4)*(1/6) = 1/24
We have 24 equally likely
pairs.
|
|
1 |
2 |
3 |
4 |
|
1 |
(1,1) |
(2,1) |
(3,1) |
(4,1) |
|
2 |
(1,2) |
(2,2) |
(3,2) |
(4,2) |
|
3 |
(1,3) |
(2,3) |
(3,3) |
(4,3) |
|
4 |
(1,4) |
(2,4) |
(3,4) |
(4,4) |
|
5 |
(1,5) |
(2,5) |
(3,5) |
(4,5) |
|
6 |
(1,6) |
(2,6) |
(3,6) |
(4,6) |
Suppose we observe the sum of the
faces in the pair of dice. Identify the possible values of this sum, and
compute (and justify)
a probability for each value.
Now for the sums:
|
|
1 |
2 |
3 |
4 |
|
1 |
(1,1) @ 2 |
(2,1) @ 3 |
(3,1) @ 4 |
(4,1) @ 5 |
|
2 |
(1,2) @ 3 |
(2,2) @ 4 |
(3,2) @ 5 |
(4,2) @ 6 |
|
3 |
(1,3) @ 4 |
(2,3) @ 5 |
(3,3) @ 6 |
(4,3) @ 7 |
|
4 |
(1,4) @ 5 |
(2,4) @ 6 |
(3,4) @ 7 |
(4,4) @ 8 |
|
5 |
(1,5) @ 6 |
(2,5) @ 7 |
(3,5) @ 8 |
(4,5) @ 9 |
|
6 |
(1,6) @ 7 |
(2,6) @ 8 |
(3,6) @ 9 |
(4,6) @ 10 |
Compute the conditional probability Pr{Sum is Even|d4 shows Even}.
Pr{ Sum is Even and d4 shows
Even } = Pr{ exactly one of (2,2), (2,4), (2,6), (4,2), (4,4), (4,6) shows } =
6/24 = 1/4 = .25
Pr{ d4 shows Even } =
Pr{ exactly one of (2,1),
(2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6)
shows } = 12/24 = 1/2 = .50
So, Pr{
Sum is Even | d4 shows Even } = .25 / .50 = .50
Continuing,…
|
|
1 |
2 |
3 |
4 |
|
1 |
(1,1) @ 2 |
(2,1) @ 3 |
(3,1) @ 4 |
(4,1) @ 5 |
|
2 |
(1,2) @ 3 |
(2,2) @ 4 |
(3,2) @ 5 |
(4,2) @ 6 |
|
3 |
(1,3) @ 4 |
(2,3) @ 5 |
(3,3) @ 6 |
(4,3) @ 7 |
|
4 |
(1,4) @ 5 |
(2,4) @ 6 |
(3,4) @ 7 |
(4,4) @ 8 |
|
5 |
(1,5) @ 6 |
(2,5) @ 7 |
(3,5) @ 8 |
(4,5) @ 9 |
|
6 |
(1,6) @ 7 |
(2,6) @ 8 |
(3,6) @ 9 |
(4,6) @ 10 |
Compute the conditional probability Pr{Sum is Odd|d6 shows Odd}.
Pr{ Sum is Odd and d6 shows
Odd } = Pr{ exactly one of (2,1), (2,3), (2,5), (4,1), (4,3), (4,5) shows } =
6/24 = 1/4 = .25
Pr{ d6 shows Odd } =
Pr{ exactly one of (1,1),
(1,3), (1,5), (2,1), (2,3), (2,5), (3,1), (3,3), (3,5), (4,1), (4,3), (4,5)
shows } = 12/24 = 1/2 = .50
So, Pr{
Sum is Odd | d6 shows Odd } = .25/.50 = 1/2 = .50
Let’s visit a few
examples from the First Hourly Deck:
HR1 – Summer Version A,
Case Three
Case Three | Color Slot Machine |
Conditional Probabilities
Here is our slot machine – on each
trial, it produces a 10-color sequence, using the table below:
|
Sequence* |
Probability |
|
RRBBRRYRRR |
.10 |
|
RRGGRGBRRB |
.10 |
|
BBYYGGYGBR |
.15 |
|
GRRGGYBRGG |
.10 |
|
BGYGYRYGYY |
.25 |
|
RRYYGRRBBY |
.10 |
|
YYGBYYBGRR |
.20 |
|
Total |
1.00 |
*B-Blue, G-Green, R-Red, Y-Yellow,
Sequence is numbered from left to right: (1st 2nd 3rd
4th 5th6th7th 8th 9th
10th )
Compute the following conditional
probabilities:
Pr{ Yellow Shows Exactly Twice | Blue Shows}
|
Sequence* |
Probability |
|
RRBBRRYRRR |
.10 |
|
RRGGRGBRRB |
.10 |
|
BBYYGGYGBR |
.15 |
|
GRRGGYBRGG |
.10 |
|
BGYGYRYGYY |
.25 |
|
RRYYGRRBBY |
.10 |
|
YYGBYYBGRR |
.20 |
|
Total |
1.00 |
Prior Probability
Pr{Blue Shows} = Pr{One of RRBBRRYRRR, RRGGRGBRRB,
BBYYGGYGBR, GRRGGYBRGG, BGYGYRYGYY, RRYYGRRBBY, YYGBYYBGRR Shows} =
Pr{RRBBRRYRRR} + Pr{RRGGRGBRRB} + Pr{BBYYGGYGBR} +
Pr{GRRGGYBRGG} + Pr{BGYGYRYGYY} + Pr{RRYYGRRBBY} + Pr{YYGBYYBGRR} =
.1+.1+.15+.1+.25+.1+.2 = 1.00
|
Sequence* |
Probability |
|
|
|
|
Total |
0 |
Joint Probability
Pr{ Yellow Shows Exactly Twice and Blue
Shows} = 0
Conditional = Joint /
Prior
Pr{ Yellow Shows Exactly Twice | Blue Shows} = Pr{
Yellow Shows Exactly Twice and Blue Shows}/Pr{Blue Shows} = 0/1 =0
Pr{ Green Shows | “BR” Shows }
|
Sequence* |
Probability |
|
RRBBRRYRRR |
.10 |
|
RRGGRGBRRB |
.10 |
|
BBYYGGYGBR |
.15 |
|
GRRGGYBRGG |
.10 |
|
Total |
.45 |
Pr{ “BR” Shows } = Pr{One of RRBBRRYRRR, RRGGRGBRRB,
BBYYGGYGBR, GRRGGYBRGG Shows} = Pr{RRBBRRYRRR}+ Pr{ RRGGRGBRRB}+
Pr{BBYYGGYGBR}+ Pr{GRRGGYBRGG} =
.1+.1+.15+.1 = .45
Pr{Green Shows and “BR” Shows}
|
Sequence* |
Probability |
|
RRGGRGBRRB |
.10 |
|
BBYYGGYGBR |
.15 |
|
GRRGGYBRGG |
.10 |
|
Total |
.35 |
Pr{ Green Shows and “BR” Shows } = Pr{One of
RRGGRGBRRB, BBYYGGYGBR, GRRGGYBRGG Shows} = Pr{ RRGGRGBRRB}+
Pr{BBYYGGYGBR}+ Pr{GRRGGYBRGG}
=.1+.15+.1 = .35
Pr{ Green Shows | “BR” Shows } = Pr{ Green Shows and
“BR” Shows }/Pr{ “BR” Shows } = .35/.45 = 7/9
Pr{ Red Shows | Green Shows}
|
Sequence* |
Probability |
|
RRGGRGBRRB |
.10 |
|
BBYYGGYGBR |
.15 |
|
GRRGGYBRGG |
.10 |
|
BGYGYRYGYY |
.25 |
|
RRYYGRRBBY |
.10 |
|
YYGBYYBGRR |
.20 |
|
Total |
.90 |
Pr{Green Shows} = Pr{One of RRGGRGBRRB, BBYYGGYGBR, GRRGGYBRGG,
BGYGYRYGYY, RRYYGRRBBY, YYGBYYBGRR Shows} =
Pr{RRBBRRYRRR} + Pr{RRGGRGBRRB} + Pr{BBYYGGYGBR} +
Pr{GRRGGYBRGG} + Pr{BGYGYRYGYY} + Pr{RRYYGRRBBY} + Pr{YYGBYYBGRR} =
.1+.15+.1+.25+.1+.2 = .90
Pr{ Red Shows and Green Shows}
|
Sequence* |
Probability |
|
RRGGRGBRRB |
.10 |
|
BBYYGGYGBR |
.15 |
|
GRRGGYBRGG |
.10 |
|
BGYGYRYGYY |
.25 |
|
RRYYGRRBBY |
.10 |
|
YYGBYYBGRR |
.20 |
|
Total |
.90 |
Pr{ Red and Green Show } = Pr{One of RRGGRGBRRB, BBYYGGYGBR,
GRRGGYBRGG, BGYGYRYGYY, RRYYGRRBBY, YYGBYYBGRR Shows} =
Pr{RRBBRRYRRR} + Pr{RRGGRGBRRB} + Pr{BBYYGGYGBR}
+ Pr{GRRGGYBRGG} + Pr{BGYGYRYGYY} + Pr{RRYYGRRBBY} + Pr{YYGBYYBGRR} =
.1+.15+.1+.25+.1+.2 = .90
Pr{ Red Shows | Green Shows} = Pr{ Red Shows and
Green Shows}/Pr{ Green Shows} = .90/.90 = 1
HR1 – Spring
2008, Case Four
Case Four: Color Slot
Machine, Computation of Conditional Probabilities
Here is our slot machine
– on each trial, it produces a 10-color sequence, using the table below:
|
Sequence* |
Probability |
|
RRBBR RYRRB |
.10 |
|
RRGGRGBRRB |
.10 |
|
BBYYRGYGBR |
.15 |
|
GRRGRGBRGB |
.10 |
|
BGYGYRYGYY |
.25 |
|
RRGYGRRBBB |
.10 |
|
YYGBYYBGRR |
.20 |
|
Total |
1.00 |
*B-Blue, G-Green, R-Red,
Y-Yellow, Sequence is numbered as 1st to 6th , from left
to right: (1st 2nd 3rd 4th 5th6th7th
8th 9th 10th )
Compute the following
conditional probabilities:
1. Pr{Red
Shows Somewhere in the 1st ─ 4th slots | Yellow
Shows Somewhere in the 7th ─ 10th slots}
Pr{Red Shows in the 1st – 4th slots|Yellow Shows in the 7th – 10th
slots} =
Pr{Red Shows in the 1st – 4th
slots and Yellow Shows in the 7th – 10th slots}/
Pr{ Yellow Shows in the 7th – 10th slots}
|
Sequence* |
Probability |
|
RRBBR RYRRB |
.10 |
|
BBYYRGYGBR |
.15 |
|
BGYGYRYGYY |
.25 |
|
Total |
0.50 |
Pr{ Yellow Shows in the 7th – 10th
slots} = Pr{One of RRBBRRYRRB, BBYYRGYGBR, BGYGYRYGYY shows} =
Pr{RRBBRRYRRB}+ Pr{BBYYRGYGBR}+ Pr{BGYGYRYGYY} =
.10+.15+.25 = .50
|
Sequence* |
Probability |
|
RRBBR RYRRB |
.10 |
|
Total |
0.10 |
Pr{ Red Shows in the 1st – 4th
slots and Yellow Shows in the 7th – 10th slots } =
Pr{One of RRBBRRYRRB shows} = .10
Pr{Red Shows in the 1st – 4th slots|Yellow Shows in the 7th – 10th
slots} = .10/.50 = .20
2. Pr{Green
Shows Anywhere | “RB” Shows Anywhere}
Pr{Green Shows Anywhere|”RB” Shows Anywhere} =
Pr{ Green Shows Anywhere and ”RB” Shows
Anywhere }/ Pr{”RB” Shows Anywhere}
|
Sequence* |
Probability |
|
RRBBR RYRRB |
.10 |
|
RRGGRGBRRB |
.10 |
|
RRGYGRRBBB |
.10 |
|
Total |
0.30 |
Pr{”RB” Shows Anywhere} = Pr{One of RRBBRRYRRB,
RRGGRGBRRB, RRGYGRRBBB Shows} = Pr{RRBBRRYRRB}+Pr{RRGGRGBRRB}+Pr{RRGYGRRBBB}
=.10+.10+.10 = .30
|
Sequence* |
Probability |
|
RRGGRGBRRB |
.10 |
|
RRGYGRRBBB |
.10 |
|
Total |
0.20 |
Pr{ Green Shows Anywhere and ”RB” Shows
Anywhere } = Pr{One of RRGGRGBRRB, RRGYGRRBBB Shows} =
Pr{RRGGRGBRRB}+Pr{RRGYGRRBBB} =.10+.10 = .20
Pr{Green Shows Anywhere|”RB” Shows Anywhere} = .20/.30
3. Pr{Yellow
Shows Anywhere | Blue Shows Anywhere}
Pr{Yellow Shows Anywhere | Blue Shows Anywhere} =
Pr{Yellow Shows Anywhere and Blue Shows
Anywhere}/Pr{ Blue Shows Anywhere}
|
Sequence* |
Probability |
|
RRBBR RYRRB |
.10 |
|
RRGGRGBRRB |
.10 |
|
BBYYRGYGBR |
.15 |
|
GRRGRGBRGB |
.10 |
|
BGYGYRYGYY |
.25 |
|
RRGYGRRBBB |
.10 |
|
YYGBYYBGRR |
.20 |
|
Total |
1.00 |
Pr{ Blue Shows Anywhere} =
Pr{one of RRBBRRYRRB, RRGGRGBRRB, BBYYRGYGBR, GRRGRGBRGB, BGYGYRYGYY,
RRGYGRRBBB, YYGBYYBGRR Shows} =Pr{RRBBRRYRRB}+Pr{RRGGRGBRRB}+Pr{ BBYYRGYGBR}+Pr{GRRGRGBRGB}+Pr{BGYGYRYGYY}+Pr{RRGYGRRBBB}+Pr{YYGBYYBGRR}
= .10+.10+.15+.10+.25+.10+.20 = 1.00
|
Sequence* |
Probability |
|
RRBBR RYRRB |
.10 |
|
BBYYRGYGBR |
.15 |
|
BGYGYRYGYY |
.25 |
|
RRGYGRRBBB |
.10 |
|
YYGBYYBGRR |
.20 |
|
Total |
0.80 |
Pr{Yellow Shows
Anywhere and Blue Shows Anywhere} = Pr{one of RRBBRRYRRB, BBYYRGYGBR,
BGYGYRYGYY, RRGYGRRBBB, YYGBYYBGRR Shows} =Pr{RRBBRRYRRB}+
Pr{BBYYRGYGBR}+Pr{BGYGYRYGYY}+Pr{RRGYGRRBBB}+Pr{YYGBYYBGRR} =
.10+.15+.25+.10+.20 = .80
Pr{Yellow Shows Anywhere | Blue Shows Anywhere} = .80/1.00 = .80
Case Study 1.13
Conditional Probability
Case Description: Compute
conditional probabilities for pairs of draws (without replacement).
Here is our bowl, in tabular form:
|
Color |
# in Bowl |
Proportion
of Bowl |
|
Blue |
5 |
5/9 |
|
Green |
3 |
3/9 |
|
Red |
1 |
1/9 |
|
Total |
9 |
1 |
Suppose that on each trial of this
experiment that we make two (2) draws without replacement from the bowl.
Compute Pr{
green shows 2nd | red shows 1st };
Here is our bowl, after "red
shows 1st", in tabular form:
|
Color |
# in Bowl – Before 1st
Draw |
#
in Bowl – After 1st Draw |
|
Blue |
5 |
5 – 0 = 5 |
|
Green |
3 |
3 – 0 = 3 |
|
Red |
1 |
1 – 1 = 0 |
|
Total |
9 |
8 |
With the red chip out of the
bowl, 3 of the 8 surviving chips are green. So, Pr{G 2nd
| R 1st} = (3-0) / (9-1) = 3/8
Compute Pr{
red shows 2nd | red shows 1st };
Pr{R 2nd | R 1st}
= (1-1) / (9-1) = 0/8 = 0. There are 1-1=0 surviving red chips after the first
draw.
Compute Pr{
blue shows 2nd | blue shows 1st }.
Here is our bowl, after "blue
shows 1st", in tabular form:
|
Color |
# in Bowl – Before 1st
Draw |
#
in Bowl – After 1st Draw |
|
Blue |
5 |
5 – 1 = 4 |
|
Green |
3 |
3 – 0 = 3 |
|
Red |
1 |
1 – 0 = 1 |
|
Total |
9 |
8 |
Pr{B 2nd | B 1st} =
(5-1)/(9-1) = 4/8. After the first draw, 4 of 8 surviving chips are blue.
HR1 – Fall
2004, Case Three
Case Three
Conditional Probability
Color Bowl/Draws without
Replacement
We have a bowl
containing the following colors and counts of balls (color@count):
Blue @ 5, Green @ 1, Red
@ 2, Yellow @ 3
Each trial of our
experiment consists of three (3) draws without replacement from the bowl.
Compute these directly.
|
Color |
Count |
|
B |
5 |
|
G |
1 |
|
R |
2 |
|
Y |
3 |
|
Total |
11 |
Pr{ green shows 2nd | green shows 1st}
|
Color |
Count |
|
B |
5 |
|
G |
1 |
|
R |
2 |
|
Y |
3 |
|
Total |
11 |
ß green shows 1st
|
Color |
Count |
|
B |
5 |
|
G |
0 |
|
R |
2 |
|
Y |
3 |
|
Total |
10 |
Pr{ green shows 2nd | green shows 1st}
= 0/10
Pr{ yellow shows 3rd | green shows 1st,
blue shows 2nd}
|
Color |
Count |
|
B |
5 |
|
G |
1 |
|
R |
2 |
|
Y |
3 |
|
Total |
11 |
ß green shows 1st
|
Color |
Count |
|
B |
5 |
|
G |
0 |
|
R |
2 |
|
Y |
3 |
|
Total |
10 |
ß blue shows 2nd
|
Color |
Count |
|
B |
4 |
|
G |
0 |
|
R |
2 |
|
Y |
3 |
|
Total |
9 |
Pr{ yellow shows 3rd | green shows 1st,
blue shows 2nd} = 3/9
Pr{ red shows 3rd | green shows 1st,
red shows 2nd }
|
Color |
Count |
|
B |
5 |
|
G |
1 |
|
R |
2 |
|
Y |
3 |
|
Total |
11 |
ß green shows 1st
|
Color |
Count |
|
B |
5 |
|
G |
0 |
|
R |
2 |
|
Y |
3 |
|
Total |
10 |
ß red shows 2nd
|
Color |
Count |
|
B |
5 |
|
G |
0 |
|
R |
1 |
|
Y |
3 |
|
Total |
9 |
Pr{ red shows 3rd | green shows 1st,
red shows 2nd } = 1/9