The probability of selecting 2 men and a woman is [tex]\frac{21}{44}[/tex]
The sample space is made up of 7 men and 5 women. So we have a total of 12 people.
If 3 applicants are randomly selected without replacement, there will be three mutually exclusive possibilities;
[tex]MMW=\text{Event that a Man, then Man, then Woman is selected}\\MWM=\text{Event that a Man, then Woman, then Man is selected}\\WMM=\text{Event that a Woman, then Man, then Man is selected}\\[/tex]
The final probability will have the form
[tex]P(\text{2 men and 1 woman})=P(MMW)+P(MWM)+P(WMM)[/tex]
because the possibilities are mutually exclusive.
Each mutually exclusive possibility is made up of dependent events. This is because when selection is done without replacement, it affects the size of the sample space.
[tex]P(MMW)=\dfrac{7}{12}\times\dfrac{6}{11}\times\dfrac{5}{10}\\\\=\dfrac{7}{44}[/tex]
[tex]P(MWM)=\dfrac{7}{12}\times\dfrac{5}{11}\times\dfrac{6}{10}\\\\=\dfrac{7}{44}[/tex]
[tex]P(MWM)=\dfrac{5}{12}\times\dfrac{7}{11}\times\dfrac{6}{10}\\\\=\dfrac{7}{44}[/tex]
The final probability is
[tex]P(\text{2 men and 1 woman})=P(MMW)+P(MWM)+P(WMM)\\\\=\dfrac{7}{44}+\dfrac{7}{44}+\dfrac{7}{44}\\\\=\dfrac{21}{44}[/tex]
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