Question

Randomly selecting four person committee consisting entirely of women from a pool of 12 men and 15 women. The event of selecting a woman and the event of selecting a woman the next time are independent or dependent? Probability of randomly selecting four person committee consisting entirely of women from a pool of 12 men and 15 women is _____?

Answer 1

We are asked if the event of selecting a woman and the event of selecting a woman the next time are independent or dependent.

We can see that events are dependent because if one woman is already picked in the first draw....the probability of choosing a woman in the second draw will be different because number of woman is reduced.

The probability of randomly selecting four person committee consisting entirely of women from a pool of 12 men and 15 women will be

`\\ =\frac{_(4)^(15)\textrm{c}}{_(4)^(27)\textrm{c}}\\ \\ =((15!)/((15-4)!4!))/((27!)/((27-4)!4!))\\ \\ =((15!)/(11!\cdot 4!))/((27!)/(23!\cdot 4!))\\ \\ =(15!\cdot 23!)/(27!\cdot 11!)`

`=(15\cdot 14\cdot 13\cdot 12\cdot 11!\cdot 23!)/(27\cdot 26\cdot 25\cdot 24\cdot 23!\cdot 11!)\\ \\ =(15\cdot 14\cdot 13\cdot 12)/(27\cdot 26\cdot 25\cdot 24\cdot)`

`=(32760)/(421200)`

`=(7)/(90)=0.07777`

Therefore, probability of randomly selecting four person committee consisting entirely of women from a pool of 12 men and 15 women is 0.0777.

Answer 2

Solution: We have to find the probability of randomly selecting four person committee consisting entirely of women from a pool of 12 men and 15 women.

The number of ways 4 women can be selected from 15 women is:

15C4
`=\binom{15}{4}=(15!)/((15-4)!4!) =(15!)/(11! * 4!) =(15* 14* 13 * 12*11!)/(11! * 4* 3 *2 *1)`

`=(15*14*13*12)/(24) =1365`

The number of ways 4 persons can be selected from total 12+15 = 27 persons is:

27C4
`=\binom{27}{4}=(27!)/((27-4)!4!)=(27!)/(23! * 4!) =(27*26*25*24*23!)/(23! * 4*3*2*1)`

`=(27*26*25*24)/(24) =17550`

Therefore, the probability of randomly selecting four person committee consisting entirely of women from a pool of 12 men and 15 women is:

`\frac{\binom{15}{4}}{\binom{27}{4}} =(1365)/(17550)=0.0778`

The event of selecting a woman and the event of selecting a woman the next time are dependent because the probability of selecting a woman at first draw is not same as the probability of selecting a woman at second draw.