Question

@Mathematics @Mathematics @Mathematics 3 students are selected at random from a class consisting of 5 freshman, 3 sophomores, and 2 juniors. Find the probability that 2 freshman and 1 junior are selected given that at least 1 freshman is selected.

Answer 1

2/11

Answer 2

please show me. i can find probability of choosing 2 freshman and 1 junior, that is 5C2 * 2C1 / ( 10 C 3 )

Answer 3

P(2f,j|at least 1 f)=P(2f,j,at least 1 f)/P(at least 1 f)

=P(2f,j)/P(at least 1 f)=P(2f,j)/(1-P(0 f))

Answer 4

wait, i dont follow

Answer 5

ok i see what you did in the denominator,

Answer 6

P(2f,j,at least 1 f) = P(2f,j) ?

Answer 7

the event {2f} is a subset of {at least one f}

so the intersection is {2f}

Answer 8

ok, so P ( 2f j & at least 1 f ) = P ( 2f j )

Answer 9

but its not a subset

Answer 10

it is

Answer 11

because you could have 3,4,.. freshman

Answer 12

{at least 1f}= {1f or 2f or 3f}

since you are picking 3

Answer 13

oh it is , { f1 , f2} subset of { f1,f2,f3,f4,f5 } ?

Answer 14

, where we dont identify f1,...

Answer 15

{2f} intersect {1f or 2f or 3f} ={2f}

Answer 16

there are 5 freshman actually

Answer 17

you are only picking 3 people though

Answer 18

so the most freshmen you can pick is 3

Answer 19

but, you could pick one freshman exactly, in which case {2f} is not subset of {1f}

Answer 20

no...you are not thinking of it in the right way

Answer 21

the event is {1f or 2f or 3f}

Answer 22

ok the whole event

Answer 23

alright, can you do this in a tree diagram ?

Answer 24

it is discrete so you can

Answer 25

by the way, you left out operation of and or

Answer 26

?

Answer 27

P(1f &2f & j|at least 1 f)=P(2f& j&at least 1 f)/P(at least 1 f)
etc
=P(2f,j)/P(at least 1 f)=P(2f,j)/(1-P(0 f))

Answer 28

you could abbreviate and say 2f = 1f & 2f

Answer 29

no

Answer 30

P(1f &2f &1 j|at least 1 f)=P(1f&2f&1 j&at least 1 f)/P(at least 1 f)

Answer 31

2f

and

1f & 2f

are two different things

Answer 32

ok what did you mean by 2f ?

Answer 33

1f & 2f can't happen

Answer 34

that you picked 2 freshmen

Answer 35

1f = choice of first freshmen, or first draw you picked a freshman
and 2f = second choice of freshman, or second draw you picked a freshman

Answer 36

I wouldn't use that notation...I would use

\[f_1, f_2\]

Answer 37

you used it before
{2f} intersect {1f or 2f or 3f} ={2f}

Answer 38

since we don't care about order anyway...I wouldn't pick them one at a time

Answer 39

no...{2f}= you picked 2 freshmen
{1f or 2f or 3f}= you pick 1 freshmen or 2 freshmen or 3 freshmen

Answer 40

so you wouldnt do , first freshman drawn, then second freshman drawn on second draw

Answer 41

no

Answer 42

im thinking of a coin? where do we use drawing, oh cards

Answer 43

first draw a heart , then draw a spade. that sort of thing

Answer 44

but then you have the issue of order does not count , so it makes it complicated

Answer 45

that is if you care about the order...here you don't

Answer 46

you have f f j, f j f , j f f

Answer 47

if you break this problem into drawings

Answer 48

sure

Answer 49

but you didnt do it that way

Answer 50

or, i mean selections

Answer 51

you can...I didn't

Answer 52

one selection at a time

Answer 53

ok so you did P ( 2f & 1j | at least 1j ) / ( P ( at least 1 j )

Answer 54

close

Answer 55

you wrote P ( 2f, j) , the comma threw me off

Answer 56

what does comma mean exactly ?

Answer 57

and

Answer 58

2f,j=2f & j

Answer 59

ok

Answer 60

but 2f = { 1f or 2f } , no it doesnt

Answer 61

ok i see

Answer 62

and P ( 2f, j) = 5C2 * 2C1 / ( 10 C3 ) ?

Answer 63

\[2f\neq1f\text{ or }2f\]
using my notation

Answer 64

yes

Answer 65

and how did you get P ( at least 1f) = 1 - P ( 0 F)

Answer 66

complement

Answer 67

since the event is choosing 3 , then we have , i dont know

Answer 68

what is P( 0f)

Answer 69

0f=pick no freshmen

Answer 70

the three students selected can be sophomores or juniors >

Answer 71

yes

Answer 72

you can lump them together

Answer 73

so it will be 5C3 / 10 c 3

Answer 74

errr, i dont know , the numerator is wrong

Answer 75

\[\frac{\frac{_5C_2\cdot_2C_1}{_{10}C_3}}{1-\frac{_5C_3}{_{10}C_3}}\]

Answer 76

so i had P(0f) right

Answer 77

yes

Answer 78

but if there are no freshman, dont we have to consider the sophomores and juniors seperately

Answer 79

we can...but it is quicker to temporally label them collectively as non-freshmen

Answer 80

ok

Answer 81

and 10 C 3 because thats the whole event

Answer 82

yes...that is the experiment ...picking 3 from 10 where order does not matter

Answer 83

how do you know that 2f,j = 2f & j, sometimes ive seen it as or

Answer 84

a , should be treated as & when inside a probability

Answer 85

not as an or. that would make the problem different if it was an 'or'

Answer 86

you can always use and if you want

Answer 87

yes...that would change the problem if it was an 'or'

Answer 88

hmmm, what would it be?
and can we do this problem over with a tree model or tree