Question

Below are listed the numbers of engineers in various fields by sex. Choose one engineer at random. MechanicalElectricalBiomedicalMale9,6014,3576,318Female2,0391,1815,168 1. Find P("Electrical") 2. Find P("Male")3. Find P ("Electrical and Male")4. Find P ("Electrical" | "Male")5. Based on 1., 2. , 3., 4. are the events "Electrical" and "Male" independent? Why?Hint: Two events A and B are independent if and only if a) P (A and B ) = P(A)P(B) orb) P (A | B) = P (A)

Answer 1

Recall the formula for the probability of an event:

`P=\frac{Number\text{ of favorable outcomes}}{\text{Total number of outcomes}}`

From the given data, the total number of engineers is:

`9601+4357+6318+2039+1181+5168=28664`

It follows that the probability P(Electrical) is:

`P(Electrical)=\frac{\text{Number of electrical engineers}}{Total\text{ number of engineers}}`

The number of electrical engineers is:

`4357+1181=5538`

Substitute into the probability:

`P(\text{Electrical)}=(5538)/(28664)=(2769)/(14332)`

The probability P(Male) from the probability formula is:

`P(\text{Male)}=\frac{\text{Number of Male engineers}}{\text{Total number of engineers}}`

The Number of male engineers is:

`9601+4357+6318=20276`

Substitute into the probability:

`P(\text{Male)}=(20276)/(28664)=(5069)/(7166)`

The probability P(Electrical and male) is:

`P(\text{Electrical and Male)}=\frac{Number\text{ of male electrical engineers}}{\text{Total number of engineers}}`

The number of electrical engineers who are male is 4357.

Hence, the probability is:

`P(\text{Electrical and male)}=(4357)/(28664)`

The formula for conditional probability is given as:

`P(A|B)=\frac{P(A\; \text{and }B)}{P(B)}`

It follows that:

`P(\text{Electrical}|\text{Male)}=\frac{P(\text{Electrical and male)}}{P(Male)}`

Substitute the calculated probabilities:

`\begin{gathered} P(\text{Electrical}|\text{male)}=((4357)/(28664))/((20276)/(28664))=(4357)/(28664)/(20276)/(28664)=(4357)/(28664)*(28664)/(20276)=(4357)/(20276) \\ \end{gathered}`

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).