Interesting little probability puzzle

[Arp220]

Apologies if this has been discussed to death already, I'm new around here /images/graemlins/wink.gif But I liked this, and so thought people here might like it too. Happened across it on teh interwebs...

Anyway:

You are given a box that contains ten marbles. You know for a fact that the marbles inside are either black or white, and that the color of each marble was chosen by tossing an unbiased coin, heads for white and tails for black. You cannot see inside the box. You pull out one marble, inspect its color, put it back inside the box, and give the box a good shake. You repeat this nine more times. If all ten marbles you pull out are white, what is the probability that all ten marbles in the box are white?

[thylacine]

[ QUOTE ]
Apologies if this has been discussed to death already, I'm new around here /images/graemlins/wink.gif But I liked this, and so thought people here might like it too. Happened across it on teh interwebs...

Anyway:

You are given a box that contains ten marbles. You know for a fact that the marbles inside are either black or white, and that the color of each marble was chosen by tossing an unbiased coin, heads for white and tails for black. You cannot see inside the box. You pull out one marble, inspect its color, put it back inside the box, and give the box a good shake. You repeat this nine more times. If all ten marbles you pull out are white, what is the probability that all ten marbles in the box are white?

[/ QUOTE ]

<font color="red"> WARNING!! WARNING!! </font> This problem is well defined and has a specific correct solution.

[Arp220]

[ QUOTE ]
[ QUOTE ]
Apologies if this has been discussed to death already, I'm new around here /images/graemlins/wink.gif But I liked this, and so thought people here might like it too. Happened across it on teh interwebs...

Anyway:

You are given a box that contains ten marbles. You know for a fact that the marbles inside are either black or white, and that the color of each marble was chosen by tossing an unbiased coin, heads for white and tails for black. You cannot see inside the box. You pull out one marble, inspect its color, put it back inside the box, and give the box a good shake. You repeat this nine more times. If all ten marbles you pull out are white, what is the probability that all ten marbles in the box are white?

[/ QUOTE ]

<font color="red"> WARNING!! WARNING!! </font> This problem is well defined and has a specific correct solution.

[/ QUOTE ]


/images/graemlins/cool.gif

[asdfasdf32]

Probability of getting w white balls in the bag

= P(W = w)

= (1/2)^10 10^Cw

where n^Cr = n! / (r! (n-r)!)


Probability of drawing 10 white balls from a bag containing w white balls

= P(D = 10 | W = w)

= (w/10)^10


Probability of a bag containing w white balls and then drawing 10 white balls from it

= P(D = 10, W = w)

= P(D = 10 | W = w) P(W = w)

= (w/10)^10 (1/2)^10 10^Cw


Probability of drawing 10 white balls from the bag

= P(D = 10)

= Sum { P(D = 10, W = w); w=0,1,2,...,10 }

= Sum { (w/10)^10 (1/2)^10 10^Cw
; w=0,1,2,...,10 }


Probability of there being 10 white balls in the bag given 10 white balls were drawn from it

= P(D = 10, W = 10) / P(D = 10)

= (1/2)^10 / Sum { (w/10)^10 (1/2)^10 10^Cw
; w=0,1,2,...,10 }

= 1 / Sum { (w/10)^10 10^Cw
; w=0,1,2,...,10 }

= 10^10 / Sum { w^10 10^Cw
; w=0,1,2,...,10 }

= 10^10 /
{ 0^10 1
+ 1^10 10
+ 2^10 45
+ 3^10 120
+ 4^10 210
+ 5^10 252
+ 6^10 210
+ 7^10 120
+ 8^10 45
+ 9^10 10
+ 10^10 1 }

= 10^10 /
{0
+ 10
+ 46080
+ 7085880
+ 220200960
+ 2460937500
+ 12697896960
+ 33897029880
+ 48318382080
+ 34867844010
+ 10000000000 }

= 10000000000 / 142469423360

= 7812500 / 111304237

= 0.07019049957640

=7%

[Arp220]

I'll have to start posting harder problems /images/graemlins/smile.gif

[rebuyboy]

50% they are either all white or they arent