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A game uses a 12-sided die which is rolled once. Each of the faces are labelled 1-12. The dice is weighted so it is four times as likely to roll a 6, and twice as likely to roll an 8, 10 or 12.

The game costs $20 to play.

If an even number is rolled, the player wins 30 dollars.

If a multiple of 3 is rolled, the player wins 40 dollars.

If any other number is rolled, they player loses the bet.

  1. What is the probability of winning:

    a. Exactly 30 dollars

    b. Exactly 40 dollars

  2. What is the expected return for each roll of the die?

I have calculated that the probability of rolling a 6 is 2/9.

The probability of rolling an 8, 10 or 12 is 1/9 each.

I do not know how to find the probability of winning EXACTLY 30 and 40 dollars, along with the expected return per roll.

Thanks!

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  • $\begingroup$ Welcome to Math SE! Please show context by including your work – have you calculated the probability that each side will come up? $\endgroup$ – Toby Mak Jun 16 at 6:20
  • $\begingroup$ Yes, I have calculated the following probabilities: Rolling a 6 has a 4/18 probability Rolling an 8, 10 or 12 has a 2/18 probability, to a total of 6/18 probability. From this we can get the probability of earning a 30 or 40 dollar prize. I am having an issue with calculating the probability of winning EXACTLY 30 and 40 dollars respectively. Thanks! $\endgroup$ – N64Kirby Jun 16 at 6:26
  • $\begingroup$ You should now put all the work you have in the question body to get a better answer. $\endgroup$ – Toby Mak Jun 16 at 6:27
  • $\begingroup$ Done, thanks for your help! $\endgroup$ – N64Kirby Jun 16 at 6:30
  • $\begingroup$ There is some ambiguity in your question: does 'four times as likely' mean four times more likely than chance (as you have calculated), or four times more likely than the other numbers? My guess is that the latter option is true, since the probabilities for the other numbers would be reduced under the first option. $\endgroup$ – Toby Mak Jun 16 at 6:35
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Hint: You have calculated the probabilities correctly. Given this, to win exactly $30$ dollars, only even numbers that are not multiples of $3$ must be rolled. In the numbers $1$ to $12$, only $2, 4, 8, 10$ satisfy the condition. To win exactly $40$ dollars, only numbers that are a multiple of $3$ but not even should be rolled. From this, you should be able to calculate the probabilities.

For the last part, the formula for the expected value is:

$$P(\text{event } 1) \cdot P(\text{expected return from event } 1) + P(\text{event } 2) \cdot P(\text{expected return from event } 2) + \cdots + P(\text{event } n) \cdot P(\text{expected return from event } n)$$

Bear in mind many of the expected returns are $0$, so this should simplify your calculation a lot.

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  • $\begingroup$ Thank you very much for your help, your explanation has helped a lot. If this has resulted in my obtaining the answer - do I mark your response as an answer? Additionally, 4/18 = 2/9 - I simplified the fractions in my post. $\endgroup$ – N64Kirby Jun 16 at 6:55
  • $\begingroup$ Thanks for your comment! Yes, you can click the green tick to accept an answer. In addition, you can always change which answer you accept if there is another answer which helps you more. $\endgroup$ – Toby Mak Jun 16 at 6:57

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