Question

There are 9 red marbles and 3 black marbles in a bag. Marbles are selected one at a time, with replacement. Each time, the color of the marble is recorded. Find the probability of selecting a red marble exactly 8 times in 10 selections.

Answer 1

since you are replacing the marble, the probability of selecting a red marble stays the same, each time you draw from the bag.

Answer 2

So would that be 9/12 is the probability or

Answer 3

Let's label choosing a red marble "S" for success. and choosing a non red marble "F" for fail.

P(S) = 9/12
P(F) = 3/12

and you want probability of
P( SSSSSSSSFF )

Answer 4

the event "SSSSSSSSFF" means you drew a red marble on the first selection, the second selection, etc, and then on the ninth selection you drew a black marble and tenth you drew a black marble

Answer 5

alright, I understand that, so what's next?

Answer 6

P(SSSSSSSSFF)
= P(S) * P(S) * P(S) * ... P(F) * P(F)
= 9/12 * 9/12 * .. 3/12 * 3/12
= (9/12) ^8 * (3/12) ^2

Answer 7

however, we are not done, because there are more ways to select 8 red marbles and 2 black marbles.
you could select 2 black marbles first, then 8 red marbles. :
FF SSSS SSSS

you could select a red marble, then a black marble, then a red marble...:
S F S F SSSS SS

Answer 8

but each arrangement of 8 S's and 2 F's has the same probability of (9/12)^8 * (3/12)^2

Answer 9

so that is the answer itself, (9/12)^8 * (3/12)^2?

Answer 10

its part of it

Answer 11

P( exactly eight S two F ) = (# ways to rearrange 8 S's,2 F's) * (9/12)^8 * (3/12)^2

Answer 12

ohhhh okay so that is all?

Answer 13

right, so you have to find the number of ways to rearrange SSSS SSSS FF.
now doing it by hand takes too long. we have a formula for it.
10! / ( 8! 2! )

Answer 14

Oh jeez haha so how do i plug that into the formula?

Answer 15

$$

\Large P( exactly~ 8 ~ red~ marbles ~ 2~black)
\\\Large = \frac{10!}{8! ~2!} \cdot \left( \frac{9}{12} \right)^8 \cdot \left(\frac{3}{12}\right)^2

$$

Answer 16

What does the "!" mean in that equation?

Answer 17

8! = 8x7x6x5x4x3x2x1
2! = 2x1

Answer 18

you can also use the expression 10 choose 8. some calculators allow you to do that
nCr on TI 83

Answer 19

Is the answer 5/254803968?

Answer 20

simple 1-10 ! table

1!=1
2!=2
3!=6
4!=24
5!=120
6!=720
7!=5040
8!=40320
9!=362880
10!=3628800

Answer 21

! means a factorial
a factorial of a numebr is a multiplication of all previous numebr exept 0 that are natural including the numebr

Answer 22

$$ \Large {
\frac{10!}{8! ~2!}\\~\\
= \frac{10 \times 9\times 8\times 7\times 6\times 5\times 4 \times 3\times 2\times 1 }{(8\times 7\times 6\times 5\times 4 \times 3\times 2\times 1)\cdot (2 \times 1 ) }


}
$$

Answer 23

but you can simplify this, notice you can cancel 8's, 7's, , etc

Answer 24

Alright I get that but I don't have a calculator on me I'm using my phone so it's not a graphic calc so i'm not sure if it calculates probability

Answer 25

$$ \Large {
\frac{10!}{8! ~2!}\\~\\
= \frac{10 \times 9\times 8\times 7\times 6\times 5\times 4 \times 3\times 2\times 1 }{(8\times 7\times 6\times 5\times 4 \times 3\times 2\times 1)\cdot (2 \times 1 ) }

\\~\\

= \frac{10 \times 9\times \color{red}{\not 8}\times \color{blue}{\not7}\times \color{violet}{\not 6}\times 5\times 4 \times 3\times 2\times 1 }{(\not \color{red}{\not 8}\times \color{blue}{\not7}\times \color{violet}{\not 6}\times 5\times 4 \times 3\times 2\times 1)\cdot (2 \times 1 ) }


}
$$

Answer 26

and so it cancels out until its 10x9/2?

Answer 27

$$ \Large {
\frac{10!}{8! ~2!}\\~\\
= \frac{10 \times 9\times 8\times 7\times 6\times 5\times 4 \times 3\times 2\times 1 }{(8\times 7\times 6\times 5\times 4 \times 3\times 2\times 1)\cdot (2 \times 1 ) }

\\~\\

= \frac{10 \times 9\times \color{red}{\not 8}\times \color{blue}{\not7}\times \color{violet}{\not 6}\times 5\times 4 \times 3\times 2\times 1 }{(\not \color{red}{\not 8}\times \color{blue}{\not7}\times \color{violet}{\not 6}\times 5\times 4 \times 3\times 2\times 1)\cdot (2 \times 1 ) }
\\~\\ \\ \therefore \\
= \frac{10\times 9 }{2\times 1}

}
$$

Answer 28

$$
\Large{ P( exactly~ 8 ~ red~ marbles ~ 2~black)
\\\Large = \frac{10!}{8! ~2!} \cdot \left( \frac{9}{12} \right)^8 \cdot \left(\frac{3}{12}\right)^2

\\ ~\\= \frac{10\times 9 }{2\times 1}\cdot \left( \frac{9}{12} \right)^8 \cdot \left(\frac{3}{12}\right)^2
\\~\\
= \frac{90}{2} \cdot \left( \frac{9}{12} \right)^8 \cdot \left(\frac{3}{12}\right)^2
\\~\\
= 45 \cdot \left( \frac{9}{12} \right)^8 \cdot \left(\frac{3}{12}\right)^2

}$$

Answer 29

So the answer is .28156? Or are there more steps?

Answer 30

you're done :P

Answer 31

thank you so much!!!!!!!!!!!!!!!!