what is the value of w1?

Question

For this problem, involving a weighted die, assume that all the odd numbers are equally likely to come up, all the even numbers are equally likely to come up, and the odd numbers are 3 times as likely to come up as the even numbers.

(1) What is the value of w1?


(2) What is the value of w4?


(3) What is the value of w2?

Answer 1

As I don't know what w(#) means, I'll approach it two ways.  Hopefully these are enough to help you solve the problem, regardless of what w stands for.

 

Let O and E be the probabilities of rolling an odd and even number, respectively.

 

Since odd numbers are three times more likely to come up than even numbers, we have P(O) = 3P(E). 

Since all probabilities must sum to 1, we have

 

P(O) + P(E) = 1

3P(E) + P(E) = 1

4P(E) = 1, or P(E) = 1/4. 

 

Since P(O) = 3P(E), we have P(O) = 3/4.

 

The probability of each event is distributed equally among all outcomes comprising the event; that is,

P(1) = P(3) = P(5) = (1/3)*(3/4) = 1/4,

and P(2)=P(4)=P(6) = (1/3)*(1/4) = 1/12.

 

Now, let X be a random variable equal to the number which comes up after the die is rolled.

 

If w4 is P(X=4), we have w4 = 1/12.

If w4 is P(X≤4), meaning the probability your roll is less than or equal to 4, we have

w4 = P(X=1) + P(X=2) + P(X=3) + P(X=4)

     =    1/4    +    1/12   +   1/4     +   1/12.

 

As long as w is a range of a random variable, this should be enough guidance to interpret it.