Sunday Times Teaser 2768 – In the Bag

by Richard England

Published: 11 October 2015 (link)

A set of snooker balls consists of fifteen reds and seven others. From my set I put some into a bag. I calculated that if I picked three balls out of the bag simultaneously at random, then there was a one in a whole-number chance that they would all be red. It was more likely that none of the three would be red — in fact there was also a one in a whole-number chance of this happening.

How many balls did I put in the bag, and how many of those were red?

5 Comments Leave one →

Brian Gladman permalink


# there must be at least three red balls and since three reds
# is less likely than none, at least four are not red 
for other in range(4, 8):
  for red in range(3, other):
    total = red + other
    t = total * (total - 1) * (total - 2)
    
    # '1 in N' odds for red must give an integer N
    c_red, rem = divmod(t, red * (red - 1) * (red - 2))
    if not rem:
      # '1 in N' odds for no red must give an integer N
      c_other, rem = divmod(t, other * (other - 1) * (other - 2))
      # odds of red must be 'longer'
      if not rem and c_red > c_other:
        print('{} total, {} red.'.format(total, red))

# there must be at least three red balls and since three reds

# is less likely than none, at least four are not red

for other in range(4, 8):

for red in range(3, other):

total = red + other

t = total * (total - 1) * (total - 2)

# '1 in N' odds for red must give an integer N

c_red, rem = divmod(t, red * (red - 1) * (red - 2))

if not rem:

# '1 in N' odds for no red must give an integer N

c_other, rem = divmod(t, other * (other - 1) * (other - 2))

# odds of red must be 'longer'

if not rem and c_red > c_other:

print('{} total, {} red.'.format(total, red))

Jim Randell permalink

At first I read the puzzle as placing 7 of the non-red balls in the bag, but that doesn’t give any solutions (and makes the question at the end a bit odd).

But if we place only some of the non-red balls in the bag then there is a single solution.


from itertools import product
from fractions import Fraction as F
from enigma import irange, multiply, printf

# probability of choosing n balls with the same quality from t
# where q of the t balls have the quality
def P(n, t, q):
  return multiply(F(q - i, t - i) for i in irange(0, n - 1))

# choose the number of non-red and red balls
for (n, r) in product(irange(3, 7), irange(3, 15)):
  # there are r red and n non-red balls
  t = r + n

  # probability of 3 being red
  P1 = P(3, t, r)

  # probability of 3 being non-red
  P2 = P(3, t, n)

  # check the conditions
  if 1 == P1.numerator == P2.numerator and P1 < P2:
    printf("n={n} r={r}, t={t}, P1={P1} P2={P2}")

from itertools import product

from fractions import Fraction as F

from enigma import irange, multiply, printf

# probability of choosing n balls with the same quality from t

# where q of the t balls have the quality

def P(n, t, q):

return multiply(F(q - i, t - i) for i in irange(0, n - 1))

# choose the number of non-red and red balls

for (n, r) in product(irange(3, 7), irange(3, 15)):

# there are r red and n non-red balls

t = r + n

# probability of 3 being red

P1 = P(3, t, r)

# probability of 3 being non-red

P2 = P(3, t, n)

# check the conditions

if 1 == P1.numerator == P2.numerator and P1 < P2:

printf("n={n} r={r}, t={t}, P1={P1} P2={P2}")

Brian Gladman permalink

Hi Jim,

I have changed the wording to (hopefully) remove the ambiguity you found.

Reply
ronan smith permalink

Is it possible to solve teasers without a computer? Most weeks, I try and fail. I have a first class degree in economics and am reasonably numerate. Maslanka’s puzzles in saturday’s Guardian are more user friendly.

Reply
- Brian Gladman permalink
  
  Hi Ronan,
  
  If your interest is in solutions without programming, you might prefer the other site I run at https://sites.google.com/site/sundaytimesteasers/. Most contributors there are solving manually and you will often find some good discussions of different approaches.
  
  Reply

Sunday Times Teaser 2768 – In the Bag

by Richard England

Published: 11 October 2015 (link)

Leave a Reply Cancel reply

Recent Posts

Recent Comments

Archives

Categories

Meta