You have five pieces of strings. You pick two ends out of the 10 at random and tie a knot. (You may tie both ends of a string together). Then you pick two ends again out of the remaining 8 and tie them. You keep doing this until there are no ends left. What is the expected value of the number of loops you will end up with? What is the expected value if there are six strings?
Cmon no one knows how to do this??
Just pretend there were 6 ropes... You want to find the expected value of the number of loops you will have when you pick two by two and tie them together.
please elaborate on what you mean by expected value.
you can gt any number of loops from 1 to the number of strings.
Expected Value means the average of the values when probabilities are taken into consideration
For example, the "expected value" when rolling a dice is
(1/6 * 1)+(1/6 * 2)+(1/6 * 3)+(1/6 * 4)+(1/6 * 5)+(1/6 * 6) = 7/2 = 3.5
Of course it is impossible to roll a 3.5 but it is the "expected value"
So for this question, calcluate the probability on getting each number of loops and multiply by the number. So, if P(x) is the probability of getting x number of loops, for 5 loops the "expected value" is
P(1)*1 + P(2)*2 + P(3)*3 + P(4)*4 +P(5)*5
the expected value varies and it can take any value from 1 to 10factorial depending upon the selection of the ends
No, you cannot get 10! loops.
There are 5 strings so the maximum amount of loops possible is 5 (If you tie both ends of each string to itself)
Basically, I want to know how to calculate the probabilty of ending up with 1 loop, 2 loops, 3 loops, 4 loops, and 5 loops.
1 loop can be obtained only in one way i.e tying all the strings together into a loop,2 loops can be obtained with each of the loops comprising the following nos of strings (1,4),(4,1),(2,3),(3,2).
3 loops can be made with(1,1,3),(1,3,1),(3,1,1),(2,2,1),(2,1,2),(1,2,2),
and so on for each number of loops from 1 to 5, where the numbers in brackets indicate the number of strings in each loop.
i suppose (1,1,3) &its permutations must be taken differently as in the case of the number of ways of obtaining a given number by rolling a die.
to find the total number of loops possible we can easily do so by applying multinomial theorem and finding the number of +ve integral solutions to
x1+x2+x3....+xn=n for each number of loops desired.