A bag contains an unknown number of red balls and yellow balls. When N balls are drawn at random (without replacement) the probability that they are all yellow is 1/2. The number of balls in the bag is the minimum for this to happen.
If the first N balls were all yellow, what is the probability that the next ball drawn is red?
Express the probability as a function of N.
clc
for red=sym(1):600
for yellow=sym(1):60
tot=red+yellow;
for n=1:10
p=prod([yellow:-1:yellow-n+1]./[tot:-1:tot-n+1]);
if p==.5
p2=(yellow-n)/(tot-n);
fprintf('%4d %4d %4d %4d %4d %4d %12.10f\n',[n,red,yellow,tot,yellow-n,tot-n,p2])
break
elseif p<.5 || isnan(p)
break
end
end
end
end
In most instances, for n>1, the probability that the next draw is yellow is (n-1)/n, making that of red be 1/n. These are the cases where the 1/2 probability is a result of the number of yellow balls being 2*n-1 and there is only one red ball.
When n is 1, there are varying probabilities for the next draw, as shown in the taable below.
There are also some sporadic cases: for example if there are 2 red and 19 yellow balls, the probability that the first 6 are all yellow is
19*18*17*16*15*14/(21*20*19*18*17*16) = 15*14/(21*20) = 1/2
so in that case of n = 6, the probability that the next selection is red will be 2/15, rather than 1/6.
The table also shows the case where there were 6 red balls with 15 yellow balls. The probability that the first two are both yellow is also 1/2 and the probability the next ball is red is 6/19, not 1/2.
n red yellow tot yellow-n tot-n prob next is yellow
1 1 1 2 0 1 0.0000000000
2 1 3 4 1 2 0.5000000000
3 1 5 6 2 3 0.6666666667
4 1 7 8 3 4 0.7500000000
5 1 9 10 4 5 0.8000000000
6 1 11 12 5 6 0.8333333333
7 1 13 14 6 7 0.8571428571
8 1 15 16 7 8 0.8750000000
9 1 17 18 8 9 0.8888888889
10 1 19 20 9 10 0.9000000000
1 2 2 4 1 3 0.3333333333
6 2 19 21 13 15 0.8666666667
1 3 3 6 2 5 0.4000000000
1 4 4 8 3 7 0.4285714286
1 5 5 10 4 9 0.4444444444
1 6 6 12 5 11 0.4545454545
2 6 15 21 13 19 0.6842105263
1 7 7 14 6 13 0.4615384615
1 8 8 16 7 15 0.4666666667
1 9 9 18 8 17 0.4705882353
1 10 10 20 9 19 0.4736842105
1 11 11 22 10 21 0.4761904762
1 12 12 24 11 23 0.4782608696
1 13 13 26 12 25 0.4800000000
1 14 14 28 13 27 0.4814814815
1 15 15 30 14 29 0.4827586207
1 16 16 32 15 31 0.4838709677
1 17 17 34 16 33 0.4848484848
1 18 18 36 17 35 0.4857142857
1 19 19 38 18 37 0.4864864865
1 20 20 40 19 39 0.4871794872
1 21 21 42 20 41 0.4878048780
1 22 22 44 21 43 0.4883720930
1 23 23 46 22 45 0.4888888889
1 24 24 48 23 47 0.4893617021
1 25 25 50 24 49 0.4897959184
Operation terminated by user during sym/normalizesym
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Posted by Charlie
on 2024-04-16 15:24:43 |
some exceptions
