A very interesting math puzzle...?

Question

Ed put some blue marbles and some red marbles into a bag. He then asked Phil to reach into the bag, without looking in, and draw out a marble. Phil drew out a blue marble. Ed asked Phil to draw out another marble and, once again, Phil drew out a blue marble.

"There must be more blue than red marbles in the bag," said Phil. "I wonder what the probability is of my drawing out another blue marble on my third try?"

Ed replied, "Exactly nine tenths of what it was of drawing a blue marble on your first draw."

Ed told Phil that he had put "ten, give or take two or three" marbles into the bag. How many marbles were in the bag at the start?


10 points goes to the correct answer with the clearest explanation!
Thank you.

Answer 1

Let's assume there were B blue and R red, for a total of N. Then on the first draw the chance is B / (B + R)

After two blues have been removed, the chance is (B - 2) / (B + R + 2)

So you have:
B / (B + R) * 0.9 = (B - 2) / (B + R - 2)

Alternatively you can write this as:
B / N * 9 / 10 = (B - 2) / (N - 2)

9B / 10N = (B - 2) / (N - 2)

Cross multiplying:
10N(B - 2) = 9B(N - 2)

10NB - 20N = 9BN - 18B
BN - 20N = -18B
(B - 20)N = -18B
N = -18B / (B - 20)

If you try values of B, you'll get a table like the following:
B --> N
5 --> 6
6 --> 7.714285714
7 --> 9.692307692
8 --> 12
9 --> 14.72727273
10 --> 18

Since you are told that you have between 7 and 13 marbles, the only possible answer is you have 12 marbles, of which 8 were blue.

At the begining your chance was 8/12 = 2/3
After removing two blues, your chance is 6/10.
2/3 x 9/10 = 18/30 = 6/10

So the answer is there were 12 marbles in the bag at the beginning (8 blue, 4 red).

Answer 2

"Exactly nine tenths of what it was of drawing a blue marble on your first draw."

So: P(1) = b/x, where b=# blue and x=# total

P(3)= (b-2)/(x-2), since two marbles have been removed and both are blue.

And: P(1) * 9/10 = p(3)

So: (b/x) * (9/10) = (b-2)/(x-2)

Noting that x must be between 7 and 13 inclusive, and b must be between 2 and x inclusive, proceed by inspection, creating a table as suugested by puzzling. The *only* solution with whole number values will be b=8 and x=12.

Answer 3

12

Answer 4

12

Answer 5

you ought to use a recursive technique. on the top, the barrel is composed of below 9 liters of water, on the grounds that some water became bumped off alongside with the wine the 2d 2 cases. the 1st removing is 3 liters, leaving you with x-3 liters of wine. the 2d removing is 3 liters of fluid with a concentration of (x-3)/x. In different words, you have: (x-3) - 3*(x-3)/x Dividing that via x supplies you your new concentration. The third removing is 3 liters of [(x-3) - 3* (x-3)/x]/x and could bypass away you with x/2 liters of wine interior the barrel or: (x-3) - 3*(x-3)/x -3*[(x-3) - 3* (x-3)/x]/x = x/2. something is a few uncomplicated algebra to simplify the equation right into a cubic formula with 3 roots. different than not so uncomplicated that i did not make an errors someplace alongside the line and land up with 15.338 liters extremely of the nicely suited quantity of (re-edit:14.542 liters - no ask your self I had a typo in my Excel formula). Microsoft Excel facilitates you to brute potential those sort of issues, on occasion with much less risk of errors (offered you enter the formula wisely and don't continuously start up with 15 liters - heh).

Answer 6

9 blue marbles and 1 red,

Answer 7

Binomial Probability Formula

A probability formula for Bernoulli trials. The probability of achieving exactly k successes in n trials is shown below.

Formula:

n = number of trials
k = number of successes
n – k = number of failures
p = probability of success in one trial
q = 1 – p = probability of failure in one trial

Example: You are taking a 10 question multiple choice test. If each question has four choices and you guess on each question, what is the probability of getting exactly 7 questions correct?

n = 10
k = 7
n – k = 3
p = 0.25 = probability of guessing the correct answer on a question
q = 0.75 = probability of guessing the wrong answer on a question