If A, B, C are any three events, then prove that P(A ∪ B ∪ C
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    Probability

    Multiple Choice QuestionsShort Answer Type

    601. If A, B are any two events, prove that:
    (a) space space space straight P left parenthesis straight A with bar on top intersection straight B with bar on top right parenthesis space equals space 1 minus straight P left parenthesis straight A union straight B right parenthesis         (b) straight P left parenthesis top enclose straight A union straight B with bar on top right parenthesis space equals space 1 minus straight P left parenthesis straight A intersection straight B right parenthesis
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    Multiple Choice QuestionsLong Answer Type

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    602.

    If A, B, C are any three events, then prove that

    P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(C ∩ A) + P(A ∩ B ∩ C)

    Let (B ∪ C) = D
    ∴  P(A ∪ B ∪ C) = P(A ∪ D) = P(A) + P(D) - P(A ∩ D)
                             = P(A) + P(B ∪ C) - P[A ∩ (B ∪ C)]
                             = P(A) + P(B) + P(C) - P(B ∩ C) - P[A ∩ (B ∪ C)]
                                                                                    [∵P(A ∪ B) = P(A) + P(B) - P(A ∩ B)]
                             =  P(A) + P(B) + P(C) - P(B ∩ C) - P[(A ∩ B) ∪ (A ∩ C)]
                                                                   [By using A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)]
                             = P(A) + P(B) + P(C) - P(B ∩ C) - {P(A ∩ B)
                                                                        + P(A ∩ C) - P[(A ∩ B) ∩ (A ∩ C]}
                             =  P(A) + P(B) + P(C) - P(B ∩ C) - P(A ∩ B)
                                                                                      - P(C ∩ A) + P(A ∩ B ∩ C)
    Hence, P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(C ∩ A) + P(A ∩ B ∩ C)

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    603. Which of the following cannot be valid assignment of probabilities for outcomes of sample space: S = {ω1, ω2, ω3, ω4, ω5, ω6, ω7}
    Assignment:
           straight omega subscript 1       <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>         straight omega subscript 3         straight omega subscript 4        straight omega subscript 5        straight omega subscript 6       space space space space straight omega subscript 7
    (a)   0.1       0.01       0.05        0.03      0.01      0.2           0.6

    (b)    1 over 7        1 over 7        1 over 7          1 over 7        1 over 7        1 over 7          1 over 7

    (c)    0.1        0.2        0.3          0.4        0.5        0.6          0.7

    (d)    -0.1      0.2         0.3          0.4       -0.2       0.1          0.3

    (e)   1 over 14     <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>      3 over 14      4 over 14       5 over 14     space 6 over 14        15 over 14
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    604. Let a sample space be S = {ω1, ω2, ω3, ω4, ω5, ω6}. Which of the following assignments of probabilities to each outcome are valid?

    Outcomes    straight omega subscript 1         <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>        straight omega subscript 3          straight omega subscript 4         straight omega subscript 5             straight omega subscript 6
    (a)              1 over 6         1 over 6         1 over 6          1 over 6          1 over 6             1 over 6

    (b)               1            0            0             0            0                0

    (c)            space space 1 over 8          2 over 3         1 third          1 third        fraction numerator negative 1 over denominator 4 end fraction           fraction numerator negative 1 over denominator 3 end fraction

    (d)            1 over 12        1 over 12        1 over 6          1 over 6          1 over 6             3 over 2

    (e)             0.1          0.2         0.3          0.4         0.5             0.6
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