Seven sportsmen (named A,B,C,D,E,F and G) buy 7 hats choosing randomly out of 4 available
colors (say G,R,W and Y).
What is the probability that all
the colors were chosen?
Using "&" to represent "and", and "|" to represent "or":
The inclusion/exclusion principle gives us:
p(g & r & w & y) = p(g)+p(r)+p(w)+p(y)
- p(g|r) - p(g|w) - p(g|y)
- p(r|w) - p(r|y) - p(w|y)
+ p(g|r|w) + p(g|r|y) + p(g|w|y) + p(r|w|y)
Now p(g), or any of the single color probabilities (that is the probability that a given color shows up among the sportsmen, not necessarily exclusively), is 1 - (3/4)^7.
The probability that a given pair of colors will have at least one representation among the sportsmen, such as p(g|r), is 1 - (1/2)^7.
And the probability of one of the triples is 1 - (1/4)^7.
And of course p(g|r|w|y) = 1, as one of these colors must be represented at least once among the sportsmen.
So the probability that all four colors will be represented is:
4 * (1 - (3/4)^7) - 6 * (1 - (1/2)^7) + 4 * (1 - (1/4)^7) - 1
which is equal to 525/1024, or about 0.5126953125.
FOR tr = 1 TO 1000000
FOR spm = 1 TO 7
choice = INT(RND(1) * 4)
used(choice) = 1
good = 1
FOR i = 0 TO 3
IF used(i) = 0 THEN good = 0
IF good THEN hits = hits + 1
PRINT tr, hits / tr, 1024 * hits / tr
finds as its last line:
1000000 .512473 524.7723
meaning that after a million trials, the fraction of successes is .512473, or, relative to a denominator of 1024: 524.7723/1024.
Posted by Charlie
on 2010-04-26 13:50:11