<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">In calculating probability, there are two rules to consider when you are determining if two events are independent or dependent and if they are mutually exclusive or not.</span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><strong>The Multiplication Rule</strong></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">If&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are two events defined on a&nbsp;<strong>sample space</strong>, then&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>B</em>)<em>P</em>(<em>A</em>|<em>B</em>).</span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">This equation can be rewritten as&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>B</em>)<em>P</em>(<em>A</em>|<em>B</em>), the multiplication rule.</span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">If&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are&nbsp;<strong>independent</strong>, then&nbsp;<em>P</em>(<em>A</em>|<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>). In this special case,&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>|<em>B</em>)<em>P</em>(<em>B</em>) becomes&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>)<em>P</em>(<em>B</em>).</span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">A bag contains four green marbles, three red marbles, and two yellow marbles. Mark draws two marbles from the bag without replacement. The probability that he draws a yellow marble and then a green marble is&nbsp;</span></span><span style="font-size:14px"><span style="color:black">P(yellow&nbsp;and&nbsp;green)=P(yellow)&sdot;P(green&nbsp;|&nbsp;yellow)<span class="math-tex">\(\frac{2}{9},\frac{4}{8}\)</span></span></span><span style="font-size:14px"><span style="color:black">. Notice that&nbsp;P(green | yellow)=<span class="math-tex">\(\frac{4}{8}\)</span></span>&nbsp;.&nbsp;</span><span style="font-size:14px"><span style="background-color:white"><span style="color:black">After the yellow marble is drawn, there are four green marbles in the bag and eight marbles in all.</span></span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><strong>The Addition Rule</strong></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:black">If&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are defined on a sample space, then&nbsp;<em>P</em>(<em>A</em>&nbsp;OR&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>B</em>) &minus;&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>).</span></span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:black">Draw one card from a standard deck of playing cards. Let&nbsp;<em>H</em>&nbsp;= the card is a heart, and let&nbsp;<em>J</em>&nbsp;= the card is a jack. These events are not mutually exclusive because a card can be both a heart and a jack.</span></span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:black">P(H&nbsp;or&nbsp;J)=P(H)+P(J)&minus;P(H&nbsp;and&nbsp;J)</span></span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:black"><span class="math-tex">\(=\frac{13}{52}+\frac{4}{52}-\frac{1}{52}\\ =\frac{16}{52} \)</span></span></span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:black">&asymp;.3077</span></span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:black">If&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are&nbsp;<strong>mutually exclusive</strong>, then&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) = 0. Then&nbsp;<em>P</em>(<em>A</em>&nbsp;OR&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>B</em>) &minus;&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) becomes <em>P</em>(<em>A</em>&nbsp;OR&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>B</em>).</span></span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:black">Draw one card from a standard deck of playing cards. Let&nbsp;<em>H</em>&nbsp;= the card is a heart and&nbsp;<em>S</em>&nbsp;= the card is a spade. These events are mutually exclusive because a card cannot be a heart and a spade at the same time. The probability that the card is a heart or a spade is</span></span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:black">P(H&nbsp;or&nbsp;S)=P(H)+P(S)=</span></span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:black"><span class="math-tex">\(=\frac{13}{52}+\frac{13}{52}\\ =\frac{26}{52}\\ =\frac{1}{2}\)</span></span></span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:black">&asymp;.3077</span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:black">If&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are&nbsp;<strong>mutually exclusive</strong>, then&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) = 0. Then&nbsp;<em>P</em>(<em>A</em>&nbsp;OR&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>B</em>) &minus;&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) becomes <em>P</em>(<em>A</em>&nbsp;OR&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>B</em>).</span></span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:black">Draw one card from a standard deck of playing cards. Let&nbsp;<em>H</em>&nbsp;= the card is a heart and&nbsp;<em>S</em>&nbsp;= the card is a spade. These events are mutually exclusive because a card cannot be a heart and a spade at the same time. The probability that the card is a heart or a spade i</span></span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:black">P(H&nbsp;or&nbsp;S)=P(H)+P(S)</span></span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:black"><span class="math-tex">\(=\frac{13}{52}+\frac{13}{52}\)</span></span></span></span></p>

<p><span style="font-size:14px">&nbsp;<span class="math-tex">\(=\frac{26}{52}=\frac{1}{2}\)</span>&nbsp; &nbsp;=.5</span></p>