The Extremes of Probability
<p><a href="https://www.prepswift.com/quizzes/quiz/prepswift-the-extremes-of-probability" target="_blank">The Extremes of Probability Exercise</a></p><p>If a problem doesn't tell us whether two events are independent or mutually exclusive, we have to consider <strong><span style="color:#8e44ad;">The Extremes of Probability</span></strong>.</p>
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none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M243.2,18.2 L242.2,215.2" stroke-dasharray="" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M456.2,18.2 L455.2,215.2" stroke-dasharray="" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g></g></g><g></g><g></g><g></g></svg> <svg height="223.6062469482422" style="width:660px;height:223.6062469482422px;font-family:Asana-Math, Asana;background:#FFF;" width="660" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g><g><g><text style="white-space:pre;stroke:none;fill:rgb(74,144,226);fill-opacity:1;font-size:15px;font-family:Arial, Helvetica, sans-serif;font-weight:400;font-style:normal;dominant-baseline:text-before-edge;text-decoration:none solid rgb(74, 144, 226);" x="62.800018310546875" y="11.79998779296875">Mutually Exclusive</text></g></g></g><g><g><g><text style="white-space:pre;stroke:none;fill:rgb(74,144,226);fill-opacity:1;font-size:15px;font-family:Arial, Helvetica, sans-serif;font-weight:400;font-style:normal;dominant-baseline:text-before-edge;text-decoration:none solid rgb(74, 144, 226);" x="305.8000183105469" y="9.79998779296875">Independent</text></g></g></g><g><g><g><text style="white-space:pre;stroke:none;fill:rgb(74,144,226);fill-opacity:1;font-size:15px;font-family:Arial, Helvetica, sans-serif;font-weight:400;font-style:normal;dominant-baseline:text-before-edge;text-decoration:none solid rgb(74, 144, 226);" x="496.8000183105469" y="9.79998779296875">Complete Overlap</text></g></g></g><g><g><g><g transform="matrix(1,0,0,1,66.80001831054688,107.39999389648438)"><path d="M567 55C567 39 552 30 524 28L480 25L480 -3C582 0 582 0 602 0C622 0 622 0 724 -3L724 25L698 28C651 35 650 35 642 80L555 705L509 705L136 112C100 56 82 40 48 32L28 27L28 -3C120 0 120 0 140 0C159 0 161 0 247 -3L247 25L195 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<p><strong><span style="color:#e74c3c;">Examples</span></strong></p>
<p style="margin-left: 40px;"><strong><u>First example</u></strong>: The probability of event $A$ is $0.4$ and the probability of event $B$ is $0.5$. What is the probability of $A$ and $B$?</p>
<p style="margin-left: 80px;">Assuming these two events are mutually exclusive, we know that they both cannot occur. So that bottom boundary is $0$. Assuming there is complete overlap, one event is completely embedded within the other. Thus, the probability they both occur is simply the smaller value, which is $0.4$. So we get...</p>
<p style="margin-left: 80px;">$$0 \leq P_{A \ and \ B} \leq 0.4$$</p>
<p style="margin-left: 40px;"><strong><u>Second example</u></strong>: The probability of event $A$ is $0.3$ and the probability of event $B$ is $0.7$. What is the probability of $A$ or $B$?</p>
<p style="margin-left: 80px;">Assuming these two events are mutually exclusive, the probability either occurs is their sum, which is $0.3 + 0.7 = 1$. If there is complete overlap, the probability either occurs is the larger value.</p>
<p style="margin-left: 80px;">$$0.7 \leq P_{A \ or \ B} \leq 1$$</p>
<p style="margin-left: 40px;"><strong><u>Third example</u></strong>: The probability of event $A$ is $0.4$ and the probability of event $B$ is $0.8$. What is the probability $A$ or $B$ occurs?</p>
<p style="margin-left: 80px;">This one is a bit trickier. Assuming they're mutually exclusive, the "or" probability would be their sum, $0.4 + 0.8 = 1.2$. But this doesn't make any sense. Probability cannot be greater than $1$. So there is SOME overlap going on here. They cannot be mutually exclusive. But it's possible for us to make as little overlap as possible and their sum equal to exactly $1$. So that's the upper boundary. How about the lower boundary? Assuming there is complete overlap, the probability of either occurring is simply the larger value, $0.8$.</p>
<p style="margin-left: 80px;">$$0.8 \leq P_{A \ or \ B} \leq 1$$</p>