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<h2 class="hd hd-2 unit-title">Probability basics</h2>
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<h2 class="hd hd-2 unit-title">Class 2 reading questions: </h2>
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Which of the following represents a valid probability table </p>
<table class="tabular" cellspacing="0" style="table-layout:auto">
<tr>
<td style="text-align:left; border:none">
(i) </td>
<td style="text-align:left; border-right:1px solid black; border:none">
outcomes </td>
<td style="text-align:left; border:none">
1 </td>
<td style="text-align:left; border:none">
2 </td>
<td style="text-align:left; border:none">
3 </td>
<td style="text-align:left; border:none">
4 </td>
</tr>
<tr>
<td style="text-align:left; border:none">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border:none">
probability </td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border:none">
1/5 </td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border:none">
1/5 </td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border:none">
1/5 </td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border:none">
1/5 </td>
</tr>
</table>
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(ii) </td>
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outcomes </td>
<td style="text-align:left; border:none">
1 </td>
<td style="text-align:left; border:none">
2 </td>
<td style="text-align:left; border:none">
3 </td>
<td style="text-align:left; border:none">
4 </td>
<td style="text-align:left; border:none">
5 </td>
</tr>
<tr>
<td style="text-align:left; border:none">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border:none">
probability </td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border:none">
1/2 </td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border:none">
1/5 </td>
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1/10 </td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border:none">
1/10 </td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border:none">
1/10 </td>
</tr>
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<p>
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A sample space has exactly two outcomes. The first has probability [mathjaxinline]p[/mathjaxinline] and the second has probability [mathjaxinline]4p[/mathjaxinline]. What is the value of [mathjaxinline]p[/mathjaxinline]? </p>
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Let [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline] be two events with [mathjaxinline]P(A) = 0.4[/mathjaxinline], [mathjaxinline]P(B) = 0.6[/mathjaxinline] and [mathjaxinline]P(A\cap B) = 0.3[/mathjaxinline]. What is [mathjaxinline]P(A \cup B)[/mathjaxinline]? </p>
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<h3 class="hd hd-3 problem-header" id="c2-rq14-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c2-rq14-problem-progress" tabindex="-1">
Problem 4
</h3>
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<p>
Our experiment consists of tossing a coin 3 times. Take the sample space to be all sequences of 3 heads or tails. </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\Omega = \{ HHH,\, HHT,\, HTH,\, HTT,\, THH,\, THT,\, TTH,\, TTT\}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
For each of the following give the size of the event. </p>
<p>
(i) The event 'there are more heads than tails'. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c2-rq14_2_1">
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<input type="radio" name="input_c2-rq14_2_1" id="input_c2-rq14_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="c2-rq14_2_1-choice_1-label" for="input_c2-rq14_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c2-rq14_2_1">
<text> 0</text>
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<text> 1</text>
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<text> 2</text>
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<div class="field">
<input type="radio" name="input_c2-rq14_2_1" id="input_c2-rq14_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="c2-rq14_2_1-choice_4-label" for="input_c2-rq14_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_c2-rq14_2_1">
<text> 3</text>
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<div class="field">
<input type="radio" name="input_c2-rq14_2_1" id="input_c2-rq14_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="c2-rq14_2_1-choice_5-label" for="input_c2-rq14_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_c2-rq14_2_1">
<text> 4</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_c2-rq14_2_1" id="input_c2-rq14_2_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="c2-rq14_2_1-choice_6-label" for="input_c2-rq14_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_c2-rq14_2_1">
<text> 5</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_c2-rq14_2_1" id="input_c2-rq14_2_1_choice_7" class="field-input input-radio" value="choice_7"/><label id="c2-rq14_2_1-choice_7-label" for="input_c2-rq14_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_c2-rq14_2_1">
<text> 6</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_c2-rq14_2_1" id="input_c2-rq14_2_1_choice_8" class="field-input input-radio" value="choice_8"/><label id="c2-rq14_2_1-choice_8-label" for="input_c2-rq14_2_1_choice_8" class="response-label field-label label-inline" aria-describedby="status_c2-rq14_2_1">
<text> 7</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_c2-rq14_2_1" id="input_c2-rq14_2_1_choice_9" class="field-input input-radio" value="choice_9"/><label id="c2-rq14_2_1-choice_9-label" for="input_c2-rq14_2_1_choice_9" class="response-label field-label label-inline" aria-describedby="status_c2-rq14_2_1">
<text> 8</text>
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<span id="answer_c2-rq14_2_1"/>
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<span class="status unanswered" id="status_c2-rq14_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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</div></div>
</p>
<p>
(ii) The event 'the first heads occurs after the first tails.' </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c2-rq14_3_1">
<fieldset aria-describedby="status_c2-rq14_3_1">
<div class="field">
<input type="radio" name="input_c2-rq14_3_1" id="input_c2-rq14_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="c2-rq14_3_1-choice_1-label" for="input_c2-rq14_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c2-rq14_3_1">
<text> 0</text>
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<div class="field">
<input type="radio" name="input_c2-rq14_3_1" id="input_c2-rq14_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="c2-rq14_3_1-choice_2-label" for="input_c2-rq14_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_c2-rq14_3_1">
<text> 1</text>
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<input type="radio" name="input_c2-rq14_3_1" id="input_c2-rq14_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="c2-rq14_3_1-choice_3-label" for="input_c2-rq14_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_c2-rq14_3_1">
<text> 2</text>
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<div class="field">
<input type="radio" name="input_c2-rq14_3_1" id="input_c2-rq14_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="c2-rq14_3_1-choice_4-label" for="input_c2-rq14_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_c2-rq14_3_1">
<text> 3</text>
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<div class="field">
<input type="radio" name="input_c2-rq14_3_1" id="input_c2-rq14_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="c2-rq14_3_1-choice_5-label" for="input_c2-rq14_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_c2-rq14_3_1">
<text> 4</text>
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<input type="radio" name="input_c2-rq14_3_1" id="input_c2-rq14_3_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="c2-rq14_3_1-choice_6-label" for="input_c2-rq14_3_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_c2-rq14_3_1">
<text> 5</text>
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<text> 6</text>
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</div>
<div class="field">
<input type="radio" name="input_c2-rq14_3_1" id="input_c2-rq14_3_1_choice_8" class="field-input input-radio" value="choice_8"/><label id="c2-rq14_3_1-choice_8-label" for="input_c2-rq14_3_1_choice_8" class="response-label field-label label-inline" aria-describedby="status_c2-rq14_3_1">
<text> 7</text>
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<div class="field">
<input type="radio" name="input_c2-rq14_3_1" id="input_c2-rq14_3_1_choice_9" class="field-input input-radio" value="choice_9"/><label id="c2-rq14_3_1-choice_9-label" for="input_c2-rq14_3_1_choice_9" class="response-label field-label label-inline" aria-describedby="status_c2-rq14_3_1">
<text> 8</text>
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<span id="answer_c2-rq14_3_1"/>
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<span class="status unanswered" id="status_c2-rq14_3_1" data-tooltip="Not yet answered.">
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