Let S be the event that a randomly chosen female aged 18-24 is a smoker. Let ...

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="&#x2229;" 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="&#x222A;" 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="(S&#x2032;)" 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;">(S′)P(S′) is given as,

<span id="MathJax-Element-7-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="(S&#x2032;)=1&#x2212(S)=1&#x2212;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;">(S′)=1−P(S)=1−0.246=0.754P(S′)=1−P(S)=1−0.246=0.754

Hence, the value of <span id="MathJax-Element-8-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="(S&#x2032;)" 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;">(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="(S&#x222A;C)=P(S)+P(C)&#x2212(S&#x2229;C)=0.246+0.830&#x2212;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;">(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=0.844

Hence, the value of <span id="MathJax-Element-10-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="P(S&#x222A;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&#x2229;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=0.2795

Hence, 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&#x2229;C&#x2032;" 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&#x2229;C&#x2032;)=P(S)&#x2212;P(S&#x2229;C)=0.246&#x2212;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=0.014

The probability of <span id="MathJax-Element-15-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="C&#x2032;" 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&#x2032;)=1&#x2212;P(C)=1&#x2212;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=0.17

The probability of S|C' is,

<span id="MathJax-Element-17-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="P(S|C&#x2032;)=P(S&#x2229;C&#x2032;)P(C&#x2032;)=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=0.0824

Hence, the value of <span id="MathJax-Element-18-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="P(S|C&#x2032;)" 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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