Math Riddle?

Question

Suppose ten marbles are inserted into a box based on the tosses of an unbiased coin, a white marble being inserted when the coin turns up heads and a black one when the coin turns up tails. Suppose someone who knows how the marbles were selected but not what their colors are selects ten marbles from the box one at a time at random, returning each marble and mixing the marbles thoroughly before making another selection. If all ten examined marbles turn out to be white, what is the probability to the nearest percent that all ten marbles in the box are white?

To the nearest percentage point. i.e..21.6543% = 22%

One Hint: Any answer under 10% is incorrect..

My answer..tell me if this makes sense
We have 10 possible outcomes here. Anywhere from 1 to 10 marbles are white, the percentages of each being the true number must add up to 100%
This goes as follows (very roughly), dont wanna run out of space.

1 White : (1/10) to the 10th power = negligible..i.e, one has a chance of 1/10 each time out of ten picks of picking the only white, so the prob would be 1/10000000000

2 White: (2/10) to the tenth = negligible

3 White: (3/10) to the tenth = negligible

4 White (4/10) to the tenth = negligible

5 White (5/10) to the tenth, same as calling heads when throwing 10 coins and having them all land heads = negligible

6 White = (6/10) to the tenth = 0.60466%

7 White = (7/10) to the tenth = 2.82475%

8 White = (8/10) to the tenth = 10.7374%

9 White = (9/10) to the tenth = 34.8678%

Add these up and subtract from 100%:

34.8678 + 10.7374 + 2.82475 + 0.60466 =~ Slightly less than 49.1%

100%-49% = 51%

That is what i came up with at least and it works. All probabilities add up to 1 (100%) More similar problems can be found on this interesting test. Some are extremely difficult. http://www.eskimo.com/~miyaguch/power.html

Answer 1

I got 7% also

(more precisely: 7.01904995763997%)

Let X(n) be the number of sets of 10 black and white marble which have n whites.

X(n) = C(10,n) = 10!/(n!*(10-n)!)

X(0) = 1 (1 set of all blacks)
X(1) = 10 (10 sets with 1 white)
X(2) = 45 (45 sets with 2 whites)
X(3) = 120
X(4) = 210
X(5) = 252
X(6) = 210
X(7) = 120
X(8) = 45
X(9) = 10
X(10) = 1 (1 set with all whites)

Let P(n) be the probability of picking all whites from a set with n white balls. This is equal to the probability of picking all whites (n/10)^10 multiplied by the number of set with n white balls X(n) divided by the total number of set (which is 2^10 or 1024)

P(n) = X(n)/1024 * (n/10)^10

P(0) = 0.0000000000000
P(1) = 0.0000000000010
P(2) = 0.0000000045000
P(3) = 0.0000006919805
P(4) = 0.0000215040000
P(5) = 0.0002403259277
P(6) = 0.0012400290000
P(7) = 0.0033102568242
P(8) = 0.0047185920000
P(9) = 0.0034050628916
P(10) = 0.0009765625000

Let W be the probability of picking all whites:
W = ∑P(n) for n = 0 to 10 = 0.0139130296250

So there is 1.39% chance of picking all white balls. If that happened, the odds Q that the balls are all white is equal to the probability of picking all whites from the sets with 10 whites out of the probability of picking all whites from any sets:

Q = P(10) / ∑P(n) = 0.0009765625000/0.0139130296250
Q = 0.0701904995763997
or
Q = 7.019%

Answer 2

The probability of the marble being put in each time is 50 percent, so to calculate the probabililty that all ten marbles will be white, you do 0.5^10 which equals .00097656

Answer 3

7.019%

Addendum: Catarthur has given an excellent analysis of this problem. In fact, in such a scenario, the greatest likelikhood is that there's 8 white marbles and 2 black in the box, with 33.92% probability, while for 9 white marbles it's 24.47%, and 7 white marbles 23.79%. Those three add up to 82.18%, leaving only 17.82% for all the other possibilities.

Answer 4

P(ALL white) = P(white) ^10= (.5)^10= .0009766 which is no where near 1%

Answer 5

I calculated exactly the same answer as Scythian's. Your assertion that the answer is over 10% is looking suspect.

Answer 6

best I can figure is 1 percent first time 50 50 to follow again 75 25 and so on or you can take each toss individual and it would be 50% each time

Answer 7

0.09765625 %