Statistics probability die question

Question

Suppose a die has been loaded so that a six is scored five times more often than any other score, while all the other scores are equally likely. Express your answers to three decimals.

I have gotten the following answers.

What is the probability of scoring a three?

0.090909091 I have deciphered since it is a 11 sided die so I simply came up with 1/11 since there is only 1/11 chance of getting a 3

What is the probability of scoring a six? 0.454545455 I have reasoned since there are 5 chances in the 11 sided die so I have gotten 5/11.

I have gotten both of them wrong. What are the answers?

Answer 1

Let $x$ be the probablity of getting a particular nonsix number. So, by question the probablity of getting a six is $5x$.

Since there are 5 nonsix numbers, the probablity of getting a nonsix number is $5x$.

Since the probablity of getting a number is $1$, the probablity of getting a six $5x$, and getting a nonsix also $5x$, so:

$$1=5x+5x$$

Solving which we get:

$$x=0.1$$

So, since the probablity of getting a particular nonsix is $x$, the probablity of getting $3$ is $0.1$.

Similarly, the probablity of getting a $6$ is $5x$, so it is $0.5$

Answer 2

Possibly the problem is in the interpretation of 'five times more often than any other score'. It can be read in two ways.

You read it as: 'the probability of getting 6 is 5 times the probability of getting 3'.

The other way of reading it (which is probably intended) is as: 'the probability of getting 6 is 5 times the probability of getting a non-6'.

In your 11 sided die the number of sides which are non-sixes is still pretty high even if only one of them reads 3.

Answer 3

A die with 5 times the probability of rolling a six is the same as a ten sided die with five sixes on it. As the sides are 1, 2, 3, 4, 5, 6, 6, 6, 6, 6.

Probability of rolling a 3 = $\frac 1{10} = 0.1$

Probability of rolling a 6 = $\frac 5{10} = 0.5$