What is the probablility?

Question

IMAGINE a circle split into 6 even equal parts. The numbers on these circles are: 1,2,3,4,5 and 6. This circle has an arrow attached to the middle.


In the above circle, a person spins the arrow. the fraction a/b is formed; where a is the # of the sector where the arrow stops after the first spin. b is the # of the sector where the arrow stops after the second spin. On every spin, each of the sectors has an equal probability on which the arrow stops. What is the probability that the fraction a/b is greater than 1?

Answer 1

There are 36 combinations of results. Write a table with 6 columns and 6 rows. Put 1...6 across the top and 1...6 down the left. The number on the top represents the "a" spin. The number on the left represent the "b" spin results. Now in the 36 boxes in the table, write >1 if a/b>1. Write =1 if a=b. Write <1 if a/b<1.

When you are done, count up the total number of >1 boxes and divide by the total number of possible combinations of results in the table, which is 36. That's the probability that a/b>1. Doing this exercise, you will find that 6 out of the 36 combinations of results will give you a/b=1. 15 will give you a/b>1 and 15 will give you a/b<1. The probability of a/b>1 occuring is 15/36, or 5/12.

Alternatively, you can think through it this way and get your answer: you know that regardless of what number the first spin is, there is 1/6 chance that the second spin will equal the first spin. So out of a total probability of 1, 1/6 will give a/b=1. So the total probability of a/b>1 and a/b<1 is 5/6. Since the two spins are independent, random, and the range of the two spins are the same, the liklihood that a/b>1 has to equal a/b<1. That tells you that probability of a/b>1 is half of 5/6. So the probability of a/b>1 is 5/12.

Answer 2

The tough way to look at this is to describe a conditional argument, i.e., given a first spin "a", what is the probability that the second spin "b" is less than "a".\

However, this is a small problem so that method isn't necessary.

There are 6 possible values for "a"
There are 6 possible values for "b"
Therefore there are 36 possible "a" then "b" combinations.

6 of these possibilities involve "a" = "b"

Of the remaining 30, half of them include a>b and the other half a
For a/b to be greater than 1, a must be greater than b


Therefore, the 15 out of 36 possibilities represent the probability that you are searching for.

15/36

If you want to see this visually, make a 6 by 6 grid of squares and put "a" across the top and "b" on the side as numbers 1 through 6. In each square, record the result of a/b > 1 as either a yes or a no. 15 of the 36 squares will have a yes.

The other answers are wrong and confusing.

Answer 3

It's the average of the probabilities of the fraction being greater than one for an initial spin of 1 (prob. of a/b>1=0), an initial spin of 2 (prob. of a/b>1 = 1/6), 3 (prob. of a/b >1 = 2/6), and so on.

So the total probability for any two spins is (0 + 1/6 + 2/6 + 3/6 + 4/6 + 5/6) / 6, which is 2.5/6 or 5/12, a probability of 0.4166666666....)

Answer 4

another way to look at the question is:
what is the probabilty that a>b will occur on the second spin.
Since there are 2 spins, the product of the probabilities is
1/6X1/6=1/36, but since both spins must be considered, then the combination probability is 2/36 or 1/18. 1 chance in 18 that a>b will hit in the second spin.