Probability Intro - Revision history

Rb at 09:59, 2 June 2022

2022-06-02T09:59:18Z

Admin at 10:33, 9 August 2013

2013-08-09T10:33:23Z

Admin: /* Probability */

2013-08-07T21:29:31Z

‎Probability

Admin: /* Probability */

2013-08-07T21:27:45Z

‎Probability

Admin: /* Probability */

2013-08-07T21:27:32Z

‎Probability

Admin: /* The addition rule of probability */

2013-08-07T21:14:49Z

‎The addition rule of probability

Admin: /* The addition rule of probability */

2013-08-07T21:12:08Z

‎The addition rule of probability

Admin: /* The addition rule of probability */

2013-08-07T21:11:46Z

‎The addition rule of probability

Admin: /* The addition rule of probability */

2013-08-07T21:11:24Z

‎The addition rule of probability

Admin at 21:09, 7 August 2013

2013-08-07T21:09:20Z

@@ Line 33: / Line 33: @@
 {| class="wikitable"
 |-
-| (a) Event <math>E</math>: A closed loop
+| (a) Event <math>E</math>: A subset
 | (b) Union: <math>E\cup F</math>
 |-

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 <li><p><math>\bar{A}\cup \bar{B}=\left( \overline{A\cap B}\right) </math></p></li></ul>
 </li></ul>
 = Footnotes =

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 * Suppose an experiment has an outcome of interest <math>E</math>. The ''relative frequency interpretation'' of probability says that assuming the experiment can be repeated a large number of times then the relative frequency of observing the outcome <math>E</math> will settle down to a ''number'', denoted <math>\Pr (E),</math> <math>P(E)</math> or Prob<math>(E),</math> called the '''probability''' of <math>E</math>.
-This is illustrated in the next Figure where the proportion of heads obtained after <math>n</math> flips of a fair coin is plotted against <math>n</math>, as <math>n</math> increases; e.g., of the first 100 flips, 46 were heads (<math>46\%</math>). Notice that the plot becomes less ‘wobbly’ after about <math>n=</math>220 and appears to be settling down to the value of <math>\frac{1}{2}</math>.
+This is illustrated in the next Figure where the proportion of heads obtained after <math>n</math> flips of a fair coin is plotted against <math>n</math>, as <math>n</math> increases; e.g., of the first 100 flips, 46 were heads (<math>46\%</math>). Notice that the plot becomes less ‘wobbly’ after about <math>n=</math>140 and appears to be settling down to the value of <math>\frac{1}{2}</math>.
 [[File:Prob_coin.jpg|frameless|600px]]

@@ Line 67: / Line 67: @@
 * Suppose an experiment has an outcome of interest <math>E</math>. The ''relative frequency interpretation'' of probability says that assuming the experiment can be repeated a large number of times then the relative frequency of observing the outcome <math>E</math> will settle down to a ''number'', denoted <math>\Pr (E),</math> <math>P(E)</math> or Prob<math>(E),</math> called the '''probability''' of <math>E</math>.
-This is illustrated in the next Figure where the proportion of heads obtained after <math>n</math> flips of a fair coin is plotted against <math>n</math>, as <math>n</math> increases; e.g., of the first 100 flips, 46 were heads (<math>55\%</math>). Notice that the plot becomes less ‘wobbly’ after about <math>n=</math>220 and appears to be settling down to the value of <math>\frac{1}{2}</math>.
+This is illustrated in the next Figure where the proportion of heads obtained after <math>n</math> flips of a fair coin is plotted against <math>n</math>, as <math>n</math> increases; e.g., of the first 100 flips, 46 were heads (<math>46\%</math>). Notice that the plot becomes less ‘wobbly’ after about <math>n=</math>220 and appears to be settling down to the value of <math>\frac{1}{2}</math>.
 [[File:Prob_coin.jpg|frameless|600px]]

@@ Line 67: / Line 67: @@
 * Suppose an experiment has an outcome of interest <math>E</math>. The ''relative frequency interpretation'' of probability says that assuming the experiment can be repeated a large number of times then the relative frequency of observing the outcome <math>E</math> will settle down to a ''number'', denoted <math>\Pr (E),</math> <math>P(E)</math> or Prob<math>(E),</math> called the '''probability''' of <math>E</math>.
-This is illustrated in the next Figure where the proportion of heads obtained after <math>n</math> flips of a fair coin is plotted against <math>n</math>, as <math>n</math> increases; e.g., of the first <math>100</math> flips, <math>55</math> were heads (<math>55\%</math>). Notice that the plot becomes less ‘wobbly’ after about <math>n=220</math> and appears to be settling down to the value of <math>\frac{1}{2}</math>.
+This is illustrated in the next Figure where the proportion of heads obtained after <math>n</math> flips of a fair coin is plotted against <math>n</math>, as <math>n</math> increases; e.g., of the first 100 flips, 46 were heads (<math>55\%</math>). Notice that the plot becomes less ‘wobbly’ after about <math>n=</math>220 and appears to be settling down to the value of <math>\frac{1}{2}</math>.
 [[File:Prob_coin.jpg|frameless|600px]]

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 <ul>
-<li><p>''Example''<math>:</math> A sample of <math>1000</math> undergraduates were asked whether they took either Mathematics, Physics or Chemistry at A-level. The following responses were obtained: <math>100</math> just took Mathematics; <math>70</math> just took Physics; <math>100</math> just took Chemistry; <math>150</math> took Mathematics and Physics, but not Chemistry; <math>40</math> took Mathematics and Chemistry, but not Physics; and, <math>240</math> took Physics and Chemistry, but not Mathematics. What proportion took all three?</p>
+<li><p>''Example'': A sample of 1000 undergraduates were asked whether they took either Mathematics, Physics or Chemistry at A-level. The following responses were obtained: 100 just took Mathematics; 70 just took Physics; 100 just took Chemistry; 150 took Mathematics and Physics, but not Chemistry; 40 took Mathematics and Chemistry, but not Physics; and, 240 took Physics and Chemistry, but not Mathematics. What proportion took all three?</p>
 <p>This can be addressed with the following diagram:</p>
 <p>[[File:Prob_Alevels.jpg|frameless|500px]]</p>

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 <math>\Pr (E\cup F)=\Pr (E)+\Pr (F)-\Pr (E\cap F).</math>
-When <math>E</math> and <math>F</math> are mutually exclusive, so that <math>E\cap F=\emptyset</math>, this rule reveals Axiom 2: <math>\Pr (E\cup F)=\Pr (E)+\Pr (F).\smallskip </math>
+When <math>E</math> and <math>F</math> are mutually exclusive, so that <math>E\cap F=\emptyset</math>, this rule reveals Axiom 2: <math>\Pr (E\cup F)=\Pr (E)+\Pr (F)</math>.
 <ul>

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 <li><p>''Example''<math>:</math> A sample of <math>1000</math> undergraduates were asked whether they took either Mathematics, Physics or Chemistry at A-level. The following responses were obtained: <math>100</math> just took Mathematics; <math>70</math> just took Physics; <math>100</math> just took Chemistry; <math>150</math> took Mathematics and Physics, but not Chemistry; <math>40</math> took Mathematics and Chemistry, but not Physics; and, <math>240</math> took Physics and Chemistry, but not Mathematics. What proportion took all three?</p>
 <p>This can be addressed with the following diagram:</p>
-<p>[[File:Prob_Alevels.jpg|frameless|600px]]</p>
+<p>[[File:Prob_Alevels.jpg|frameless|500px]]</p>
 <p>The shaded area contains the number who took all three, which can be deduced from the above information (since the total of the numbers assigned to each part of the Venn diagram must be <math>1000</math>). The answer is therefore <math>30\%</math> (being <math>300</math> out of <math>1000</math>).</p></li>
 <li><p>Two further results on unions, intersections and complements which are of use (and which are fairly easy to demonstrate using Venn diagrams) are '''de Morgan Laws''':</p>

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 the union of <math>3</math> mutually exclusive events. These mutually exclusive events are depicted by the shaded areas <math>\mathbf{a,}</math> <math>\mathbf{b}</math> and <math>\mathbf{c}</math>, respectively, in the next Figure.
-[[File:Prob_add.jpg|frameless|600px]]
+[[File:Prob_add.jpg|frameless|500px]]
 Then by Axiom <math>3</math>, and from the fact that the three events <math>\left( E\cap\bar{F}\right) </math>, <math>\left( E\cap F\right) </math> and <math>\left( \bar{E}\cap F\right)</math> are mutually exclusive so that the “area” occupied by <math>E\cup F</math> is simply <math>\mathbf{a+b+c,}</math>

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 &&-\Pr (E_{1}\cap E_{2})-\Pr (E_{1}\cap E_{3})-\Pr (E_{2}\cap E_{3}) \\
 &&+\Pr (E_{1}\cap E_{2}\cap E_{3}).\end{aligned}</math>
 = Footnotes =