Brain Teaser
Buford bought himself a new circular saw from Sears. Unable to control his desire to cut something, he decides to try it out on Festus' pool cue. Buford cuts the pool cue into 3 random pieces.
What is the probability that Buford can make a triangle out of the 3 pieces of Festus' pool cue?
Answer
The probability, that a triangle can be made by randomly dividing a pool cue into 3 parts, is 25%
A triangle can be made, if and only if, sum of two sides is greater than the third side. Thus,
X1 < X2 + X3
X2 < X3 + X1
X3 < X1 + X2
We'll assume that the length of the pool cue is 1.
Therefore, X1 + X2 + X3 = 1
From above equations: X1 < 1/2, X2 < 1/2, X3 < 1/2
Thus, a triangle can be formed, if all three sides are less than 1/2 and sum is 1.
Now, let's find the probability that one of X1, X2, X3 is greater than or equal to 1/2.
Note that to divide pool cue randomly into 3 parts, we need to choose two numbers P and Q, both are between 0 & 1 and P < Q. Thus, X1 = P, X2 = Q-P, X3 = 1-Q
Now, X1 will be greater than or equal to 1/2, if and only if both the numbers, P & Q, are greater than or equal to 1/2. Thus, probability of X1 being greater than or equal to 1/2 is = (1/2) * (1/2) = 1/4
Similarly, X3 will be greater than or equal to 1/2, if and only if both the numbers, P & Q, are less than or equal to 1/2. Thus, probability of X3 being greater than or equal to 1/2 is = (1/2) * (1/2) = 1/4
Also, probability of X2 being greater than or equal to 1/2 is = (1/2) * (1/2) = 1/4
The probability that a triangle can not be made
= (1/4) + (1/4) + (1/4)
= (3/4)
Thus, the probability that a triangle can be made
= 1 - (3/4)
= (1/4)
= 25 %
Thus, the probability that a triangle can be made by randomly dividing a stick of length 1 into 3 parts is 25%