Triangle Probability

Flying Distance: Flying Distance

Flying Distance: Flying Distance

posted by padmaram @ 12:04 AM 0 comments

Flying Distance: Flying Distance

Flying Distance: Flying Distance

posted by padmaram @ 12:04 AM 0 comments

Triangle Probability

A stick of length 1 is divided randomly into 3 parts. What is the probability that a triangle can be made with those three parts?

Answer:

The probability, that a triangle can be made by randomly dividing a stick of length 1 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 <>
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 stick randomly into 3 parts, we need to choose two numbers P and Q, both are between 0 & 1 and P. 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%.

ADDITIONAL INFORMATION FOR INTERESTED PUZZLE LOVERS:

Let's generalize the problem. What is the probability that a polygon with (N+1) sides can be made from (N+1) segments obtained by randomly dividing a stick of length l into (N+1) parts?The probability is = 1 - (N+1)*(1/2)^N. The probability tends to 1 as N grows. Thus, it is easier to make a N-sided polygon than it is to make a triangle!!!