Straight Lines

Question
CBSEENMA11014643

From a group of 2 boys and 3 girls, two children selected at random. Describe the events:

A : Both children are girls.

B : The selected group consists of one boy and one girl.

C : The group consists of atleast one girl.

Find: (a) Events: B', C - A, A ∪ B,B ∩ C.

(b) Which pairs of events are mutually exclusive?

Solution

Let the boys be straight B subscript 1 comma space straight B subscript 2 and the girls be space straight G subscript 1 comma space straight G subscript 2 comma space straight G subscript 3
Two children are selected.
∴ The sample space is:
                           <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>
A: Both children are girls.
rightwards double arrow                        straight A equals open curly brackets straight G subscript 1 space straight G subscript 2 comma space straight G subscript 1 space straight G subscript 3 comma space straight G subscript 2 space straight G subscript 3 close curly brackets
B: Selected group consists of one boy and one girl.
rightwards double arrow                       straight B equals open curly brackets straight B subscript 1 space straight G subscript 1 comma space straight B subscript 1 space straight G subscript 2 comma space straight B subscript 1 space straight G subscript 3 comma space straight B subscript 2 space straight G subscript 1 comma space straight B subscript 2 space straight G subscript 2 comma space straight B subscript 2 space straight G subscript 3 close curly brackets
C: Selected group consists of atleast one girl.
rightwards double arrow                       straight C equals open curly brackets straight B subscript 1 straight G subscript 1 comma space straight B subscript 1 straight G subscript 2 comma space straight B subscript 1 straight G subscript 3 comma space straight B subscript 2 straight G subscript 1 comma space straight B subscript 2 straight G subscript 2 comma space straight B subscript 2 straight G subscript 3 comma space straight G subscript 1 straight G subscript 2 comma space straight G subscript 1 straight G subscript 3 comma space straight G subscript 2 straight G subscript 3 close curly brackets
(a)                       straight B apostrophe space equals space straight S minus straight B space equals space left curly bracket straight B subscript 1 space straight B subscript 2 comma space straight G subscript 1 space straight G subscript 2 comma space straight G subscript 1 space straight G subscript 3 comma space straight G subscript 2 space straight G subscript 3 right curly bracket
                  straight C minus straight A equals open curly brackets straight B subscript 1 straight G subscript 1 comma space straight B subscript 1 straight G subscript 2 comma space straight B subscript 1 straight G subscript 3 comma space straight B subscript 2 straight G subscript 1 comma space straight B subscript 2 straight G subscript 2 comma space straight B subscript 2 straight G subscript 3 comma space straight G subscript 1 straight G subscript 2 comma space straight G subscript 1 straight G subscript 3 comma space straight G subscript 2 straight G subscript 3 space close curly brackets
                                                                      negative open curly brackets straight G subscript 1 space straight G subscript 2 comma space straight G subscript 1 space straight G subscript 3 comma space straight G subscript 2 space straight G subscript 3 close curly brackets
                            = open curly brackets straight B subscript 1 straight G subscript 1 comma space straight B subscript 1 straight G subscript 2 comma space straight B subscript 1 straight G subscript 3 comma space straight B subscript 2 straight G subscript 1 comma space straight B subscript 2 straight G subscript 2 comma space straight B subscript 2 straight G subscript 3 close curly brackets
                  straight A union straight B space equals space left curly bracket straight G subscript 1 straight G subscript 2 comma space straight G subscript 1 straight G subscript 3 comma space straight G subscript 2 straight G subscript 3 right curly bracket space union space open curly brackets straight B subscript 1 straight G subscript 1 comma space straight B subscript 1 straight G subscript 2 comma space straight B subscript 1 straight G subscript 3 comma space straight B subscript 2 straight G subscript 1 comma space straight B subscript 2 straight G subscript 2 comma space straight B subscript 2 straight G subscript 3 close curly brackets
                         <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>
                  space space space space straight B intersection straight C space equals space open curly brackets straight B subscript 1 straight G subscript 1 comma space straight B subscript 1 straight G subscript 2 comma space straight B subscript 1 straight G subscript 3 comma space straight B subscript 2 straight G subscript 1 comma space straight B subscript 2 straight G subscript 2 comma space straight B subscript 2 straight G subscript 3 close curly brackets
                           intersection space open curly brackets straight B subscript 1 straight G subscript 1 comma space straight B subscript 1 straight G subscript 2 comma space straight B subscript 1 straight G subscript 3 comma space straight B subscript 2 straight G subscript 1 comma space straight B subscript 2 straight G subscript 2 comma space straight B subscript 2 straight G subscript 3 comma space straight G subscript 1 straight G subscript 2 comma space straight G subscript 1 straight G subscript 3 comma space straight G subscript 2 straight G subscript 3 close curly brackets
                            <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> open curly brackets straight B subscript 1 straight G subscript 1 comma space straight B subscript 1 straight G subscript 2 comma space straight B subscript 1 straight G subscript 3 comma space straight B subscript 2 straight G subscript 1 comma space straight B subscript 2 straight G subscript 2 comma space straight B subscript 2 straight G subscript 3 close curly brackets equals straight B.
rightwards double arrow             <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>
(b)              <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> A and B are mutually exclusive.
                    straight A intersection straight C space equals space open curly brackets straight G subscript 1 straight G subscript 2 comma space straight G subscript 1 straight G subscript 3 comma space straight G subscript 2 straight G subscript 3 close curly brackets space not equal to straight ϕ.
                    straight B intersection straight C space equals space straight B not equal to straight ϕ.
Hence, only events A and B are mutually exclusive.