Let S be the event that a randomly chosen female aged 18-24 is a smoker. Let ...
本帖最后由 奥密克龙 于 2022-4-13 08:14 编辑{:2_25:}Question:Let S be the event that a randomly chosen female aged 18-24 is a smoker. Let C be the event that a randomly chosen female aged 18-24 is a Caucasian. Given P(S) = .246, P(C) = .830, and P(S <span id="MathJax-Element-1-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="∩" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.8px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">∩∩ C) = .232, find each probability.a. P(S').b. P(S <span id="MathJax-Element-2-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="∪" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.8px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">∪∪ C).c. P(S | C).d. P(S | C').
Conditional Probability:
The concept of conditional probability is the one that can be used to obtain the likelihood value (probability) that how much an event is dependent other events happening, that is if one event has occurred how much the occurrence of other event is affected.
Answer and Explanation:
Given Information
[*]S be the event that female (18-24 age) chosen is a smoker.
[*]C be the event that female (18-24 age) chosen is a Caucasian.
[*]P(S)=0.246P(S)=0.246
[*]P(C)=0.830P(C)=0.830
[*]P(S∩C)=0.232P(S∩C)=0.232
a.The value of <span id="MathJax-Element-6-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="P(S′)" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.8px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">P(S′)P(S′) is given as,<span id="MathJax-Element-7-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="P(S′)=1−P(S)=1−0.246=0.754" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.8px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">P(S′)=1−P(S)=1−0.246=0.754P(S′)=1−P(S)=1−0.246=0.754Hence, the value of <span id="MathJax-Element-8-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="P(S′)" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.8px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">P(S′)P(S′) is found to be 0.754.
b.Using the addition law of probability,<span id="MathJax-Element-9-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="P(S∪C)=P(S)+P(C)−P(S∩C)=0.246+0.830−0.232=0.844" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.94px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">P(S∪C)=P(S)+P(C)−P(S∩C)=0.246+0.830−0.232=0.844P(S∪C)=P(S)+P(C)−P(S∩C)=0.246+0.830−0.232=0.844Hence, the value of <span id="MathJax-Element-10-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="P(S∪C)" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.94px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">P(S∪C)P(S∪C) is 0.844.
c.The probability of S|C is,<span id="MathJax-Element-11-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="P(S|C)=P(S∩C)P(C)=0.2320.830=0.2795" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.8px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">P(S|C)=P(S∩C)P(C)=0.2320.830=0.2795P(S|C)=P(S∩C)P(C)=0.2320.830=0.2795Hence, the value of <span id="MathJax-Element-12-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="P(S|C)" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.8px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">P(S|C)P(S|C) is 0.2795.
d.The probability of <span id="MathJax-Element-13-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="S∩C′" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.94px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">S∩C′S∩C′ is,<span id="MathJax-Element-14-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="P(S∩C′)=P(S)−P(S∩C)=0.246−0.232=0.014" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.94px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">P(S∩C′)=P(S)−P(S∩C)=0.246−0.232=0.014P(S∩C′)=P(S)−P(S∩C)=0.246−0.232=0.014The probability of <span id="MathJax-Element-15-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="C′" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.94px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">C′C′ is,<span id="MathJax-Element-16-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="P(C′)=1−P(C)=1−0.830=0.17" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.94px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">P(C′)=1−P(C)=1−0.830=0.17P(C′)=1−P(C)=1−0.830=0.17The probability of S|C' is,<span id="MathJax-Element-17-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="P(S|C′)=P(S∩C′)P(C′)=0.0140.17=0.0824" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.94px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">P(S|C′)=P(S∩C′)P(C′)=0.0140.17=0.0824P(S|C′)=P(S∩C′)P(C′)=0.0140.17=0.0824Hence, the value of <span id="MathJax-Element-18-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="P(S|C′)" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.94px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">P(S|C′)P(S|C′) is 0.0824.
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