Question

Cards are drawn with replacement from a standard shuffled deck repeatedly until a black 8 appears
1) What is the probability of the game stopping on exactly the 5 th card?
2) What is the probability of the game lasting at least 17 cards draws?

Answer 1

what game

Answer 2

is not a game is it just a geomtric density

Answer 3

a bit complec

Answer 4

First one is a Bernoulli trial of the form:\[\large \left(\begin{matrix}5 \\ 1\end{matrix}\right)(\frac{1}{52})^1(\frac{51}{52})^4\times\frac{1}{5}\]

Answer 5

k

Answer 6

The second one is extremely tedious using bernoulli trials so there has to be a better way

Answer 7

i see

Answer 8

a)
(50/52) ^4 * 2/52

Answer 9

where did the 4 come from ?

Answer 10

since we want probability of
P( F F F F S )
where F = non black 8
S = black 8

Answer 11

F stands for failure, S for success

Answer 12

ok

Answer 13

4th is success and 1 is failure

Answer 14

also why is 2/52?

Answer 15

i just wonder

Answer 16

there are two black 8 cards

Answer 17

http://www.jfitz.com/cards/classic-playing-cards.png

Answer 18

i see

Answer 19

those 52 cards, 2 are black 8 i see

Answer 20

i guess we still need to figure out the second one

Answer 21

let X = number of cards till you drawing a black 8
you want
P( X >= 17) = 1 - P( X < 17)

Answer 22

k

Answer 23

let X = number of cards till you draw a black 8
you want P( X >= 17)
by complement rule:
P( X >= 17) = 1 - P( X <17) = 1 - P( X<= 16)

Answer 24

IN that case, I think the second one would be:\[\large 1-\sum_{n=0}^{15}(\frac{50}{52})^n(\frac{2}{52})\]

Answer 25

does that mean n is 0 or 17?

Answer 26

n is like an index
n=0,1,2,3...15

Answer 27

that fills cases from 1 to 16, inclusive. But n is just the index of summation like @jayzdd said. The Summation can be done with a graphing calculator or online.

Answer 28

that going to be a huge calculation

Answer 29

not for a computer

Answer 30

http://www.wolframalpha.com/input/?i=1-+sum%28n%3D0..15%29+%2850%2F52%29^n+*+2%2F52

Answer 31

ya i know but using wolfram is going to take me while to find the command

Answer 32

Here: This is your final answer
http://www.wolframalpha.com/input/?i=1-%28sum+from+n%3D0+to+15%2C+%2850%2F52%29%5En%282%2F52%29%29

Answer 33

wolfram is user friendly, you dont need to know the command.
i have used trial and error.
a few times i have had to look up a command, for sophisticated problems

Answer 34

alright thank you for showing me the command on wolfram

Answer 35

i just google it for command lol

Answer 36

no thats not the command

Answer 37

we are using plain english

Answer 38

btw is 16 not 15

Answer 39

wolfram uses mathematica language, which has special syntax
or it can use english if its precise enough to understand it

Answer 40

but anywhow thank so much for clarification

Answer 41

its 15 because I start at 0 for a specific reason

Answer 42

isn't the "lasting at least 17 cards"

Answer 43

at least means greater than or equal to 17, so I set up the sum to start at 0 and head up to 15, which functionally translates as 16 because of the corresponding Bernoulli Trial

Answer 44

that make more sense thank you