Question

The scales forces of a business consists of 10 men and 10 women. A production unit of 7 people is set up at random. What is the probability that will consists of 5 men and 2 women?
The probability is?

Answer 1

I need just the setup of how to get the answer

Answer 2

first, state the probability of getting 5 men out of 20 total people
then state the probability of getting 2 women out of 20 total people
since we're trying to find the probability of 5 men AND 2 women, "AND" in probability language is a multiplication

Answer 3

(5c10+2c10)/7c20

Answer 4

Is that right

Answer 5

the c part is throwing me off

Answer 6

are they asking you for a formula? I don't do formulas, but I can tell you how to go about finding the answer

Answer 7

I just need to be shown how to get the answer. I'm not sure. I'm just guessing on what I've seen.

Answer 8

Ok let's walk through it

Answer 9

since there are 10 men and 10 women (20 people total), what is the probability of picking a woman out of the bunch?

Answer 10

going at it one by one helps you with the big picture

Answer 11

2

Answer 12

no. there are 10 women, 10 men. the probability of picking a woman is the number of women divided by the total number of people = 10/20 = 1/2

Answer 13

get it?

Answer 14

Ok I see

Answer 15

alright so after i pick the first woman, out of 20 people, you have 19 left. 9 of them are women, 10 of them are men

Answer 16

so the probability of picking the second woman is the number of women left divided by the number of people left

Answer 17

what would that be?

Answer 18

19/20

Answer 19

nope. how many women are left?

Answer 20

(after picking 1 woman out)

Answer 21

9

Answer 22

good

Answer 23

how many total people are left?

Answer 24

(after picking 1 woman out)

Answer 25

19

Answer 26

great

Answer 27

since there are only 9 women left, out of a total 19 people left, the probability of picking the 2nd woman = 9/19

Answer 28

get it?

Answer 29

yes so the probability is 9/19

Answer 30

based on what we did, what is the probability of picking the 3rd woman?

Answer 31

(we're going to end up picking 5 women and 2 men, so we're not done)

Answer 32

what is the probability of picking the 3rd woman?

Answer 33

9/18

Answer 34

no

Answer 35

the total number of people left is correct (18), but the number of women left is not correct

Answer 36

how many women do we have left after taking out 2 women?

Answer 37

7

Answer 38

um, we started with 10 women and we took out 2. how many is left?

Answer 39

8

Answer 40

yes, the probability of picking the 3rd woman is 8 (the number of women left) divided by 18 (the total number of people left) = 8/18

Answer 41

so what is the probability of picking the 4th woman?

Answer 42

(we have taken out 3 women)

Answer 43

7

Answer 44

7 out of?

Answer 45

17

Answer 46

good

Answer 47

and the probability of picking the 5th woman?

Answer 48

6/16

Answer 49

great. now we're done with the women. on to the men.

Answer 50

we have 10 men left, and since we've taken out 5 women, we have a total of 15 people left

Answer 51

so the probability of picking the first man = 10/15

Answer 52

what is the probability of picking the 2nd man?

Answer 53

9/14

Answer 54

that's correct. so now we know the probability of picking each of the 5 women and 2 men, now let's combine them.

Answer 55

if you have to pick 5 women AND 2 men, then we get to multiply the probability of picking each of them

Answer 56

so the probability: 10/20 x 9/19 x 8/18 x 7/17 x 6/16 x 10/15 x 9/14

Answer 57

so multiply all that together

Answer 58

yes

Answer 59

I'm not getting how to multiply all that together with fractions. Do u divided the fractions as low as they can go

Answer 60

i hate to butt in, but there is a different way of doing this, that might make some sense.

Answer 61

Can u show me please

Answer 62

i will write it and the reasoning behind it, and you can see which method you prefer
you are going to pick 7 out of 20
the number of ways to pick 7 out of twenty is called "20 choose 7" written in notation as
\[\dbinom{20}{7}\] and computed via
\[\dbinom{20}{7}=\frac{20\times 19\times 18\times 17\times 16\times 15\times 14}{7\times 6\times 5\times 4\times 3\times 2}\]
that will be your denominator
cancel first, multiply last or use a calculator

Answer 63

now of those 7 you want 5 men and 2 women
the number of ways to choose 5 men out of 10 is also called "10 choose 5" written as \[\dbinom{10}{5}\] and computed via
\[\frac{10\times 9\times 8\times 7\times 6}{5\times 4\times 3\times 2}\] similarly the number of ways to choose 2 out of 10 women is
\[\dbinom{10}{2}=\frac{10\times 9}{2}\]

Answer 64

you multiply these numbers together by the counting principle, and put them in the numerator
so the very short answer to your equation is
\[\frac{\dbinom{10}{5}\times \dbinom{10}{2}}{\dbinom{20}{7}}\]

Answer 65

*question, not equation

Answer 66

writing
\[\frac{\dbinom{10}{5}\times \dbinom{10}{2}}{\dbinom{20}{7}}\] really just restates the question
the numerator is the number of ways to choose 5 out of 10, times the number of ways to choose 2 out of ten
the denominator is the number of ways to choose 7 out of 20

Answer 67

some calculators have buttons for these, so you can do it that way
otherwise you have a ton of cancelling and multiplication to do

Answer 68

if it was me, i would use wolfram

Answer 69

what is that

Answer 70

10/20/7

Answer 71

how do u divide that when 20/7 want divide down

Answer 72

unfortunately, the very well explained answer given above by @the_lefay is missing a piece
you would also need to multiply by another factor to take in to account permutations

Answer 73

@mariapernez11 do you know how to compute \(\dbinom{n}{k}\) in general?

Answer 74

no I don't

Answer 75

then you need to learn that to do these problems,
that should come first

Answer 76

\[\left(\begin{matrix}n \\ k\end{matrix}\right)=_nC_k=\frac{n!}{(n-k)!k!}\]And k! = k(k-1)(k-2)...(3)(2)(1)
n! = n(n-1)(n-2)...(3)(2)(1)
(n-k)! = (n-k)(n-k-1)(n-k-2)...(3)(2)(1)
Perhaps.

Answer 77

I have two more of these two do. What is the answer to this one, cause I'm so confused on this problem

Answer 78

I'll use that formula on the next two

Answer 79

here lets ask wolfram to compute the answer to the first one
http://www.wolframalpha.com/input/?i=%28%2810+choose+2%29*%2810+choose+5%29%29%2F%2820+choose+7%29

Answer 80

@satellite73 thanks for butting in and introducing wolfram alpha! Hadn't heard of it before and it's wicked cool!

Answer 81

@the_lefay yea for a problem like this, just type it in and out it pops
who wants to compute all that nonsense