Recall the formula for the probability of an event:
[tex]P=\frac{Number\text{ of favorable outcomes}}{\text{Total number of outcomes}}[/tex]From the given data, the total number of engineers is:
[tex]9601+4357+6318+2039+1181+5168=28664[/tex]It follows that the probability P(Electrical) is:
[tex]P(Electrical)=\frac{\text{Number of electrical engineers}}{Total\text{ number of engineers}}[/tex]The number of electrical engineers is:
[tex]4357+1181=5538[/tex]Substitute into the probability:
[tex]P(\text{Electrical)}=\frac{5538}{28664}=\frac{2769}{14332}[/tex]The probability P(Male) from the probability formula is:
[tex]P(\text{Male)}=\frac{\text{Number of Male engineers}}{\text{Total number of engineers}}[/tex]The Number of male engineers is:
[tex]9601+4357+6318=20276[/tex]Substitute into the probability:
[tex]P(\text{Male)}=\frac{20276}{28664}=\frac{5069}{7166}[/tex]The probability P(Electrical and male) is:
[tex]P(\text{Electrical and Male)}=\frac{Number\text{ of male electrical engineers}}{\text{Total number of engineers}}[/tex]The number of electrical engineers who are male is 4357.
Hence, the probability is:
[tex]P(\text{Electrical and male)}=\frac{4357}{28664}[/tex]The formula for conditional probability is given as:
[tex]P(A|B)=\frac{P(A\; \text{and }B)}{P(B)}[/tex]It follows that:
[tex]P(\text{Electrical}|\text{Male)}=\frac{P(\text{Electrical and male)}}{P(Male)}[/tex]Substitute the calculated probabilities:
[tex]\begin{gathered} P(\text{Electrical}|\text{male)}=\frac{\frac{4357}{28664}}{\frac{20276}{28664}}=\frac{4357}{28664}\div\frac{20276}{28664}=\frac{4357}{28664}\times\frac{28664}{20276}=\frac{4357}{20276} \\ \end{gathered}[/tex]Based on the calculated probabilities in 1-4, it can be seen that the probabilities P(Electrical and male) ≠ P(Electrical) P(Male), also P(Electrical | male) ≠ P(electrical).
From the definitions given in the hint, it can be inferred that the events "Electrical" and "Male" are not independent.
The events "Electrical" and "Male" are not independent, because:
P(Electrical and male) ≠ P(Electrical) P(Male) and P(Electrical | male) ≠ P(electrical).
