Science:Math Exam Resources/Courses/MATH103/April 2012/Question 02 (e)

MATH103 April 2012

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Question 02 (e)

Short answer question. Show in detail how you arrive at your answer (work will be considered for this problem). In (a simplified version of) the dice game Pass the Pigs two model pigs serve as dice. Each model pig can either land on the side (with probability p(S) = 4/8), on the feet (with probability p(F) = 3/8), or on the snout (with probability p(N) = 1/8). In each turn both model pigs are rolled. The scoring is as follows: pig 1 side pig 1 feet pig 1 snout pig 2 side 0 1 5 pig 2 feet 1 3 7 pig 2 snout 5 7 10 Find the average score per turn. You can leave your answer unsimplified as a sum of fractions.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Remember that the average number of points should be ${\begin{aligned}0p(X=0)+1p(X=1)+3p(X=3)+5p(X=5)+7p(X=7)+10p(X=10),\end{aligned}}$ where $X$ is a random variable representing the number of points. Find the probabilities that X is equal to each value, and compute the sum.

Hint 2

We can find the probability that $X$ is equal to a given value by looking at all of the different combinations of rolls that can give a value for $X$. For example: ${\begin{aligned}p(X=1)&=p({\text{pig 1 lands on its side and pig 2 lands on its feet}})\\&+p({\text{pig 2 lands on its side and pig 1 lands on its feet}})\\p(X=1)&=\left({\frac {3}{8}}\right)\left({\frac {4}{8}}\right)+\left({\frac {3}{8}}\right)\left({\frac {4}{8}}\right)={\frac {24}{64}}.\end{aligned}}$ Now, perform the same calculation for the other possible values of $X$.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. This game has 9 outcomes, but due to symmetry only 6 of them are distinct. The probabilities are distributed as follows (multiples of 1/64) pig 1 side pig 1 feet pig 1 snout pig 2 side 4 × 4 4 × 3 4 × 1 pig 2 feet 3 × 4 3 × 3 3 × 1 pig 2 snout 1 × 4 1 × 3 1 × 1 Let X be the random variable that returns the score. Then ${\begin{aligned}p(X=0)&={\frac {16}{64}}&p(X=1)&=2{\frac {12}{64}}&p(X=5)&=2{\frac {4}{64}}\\p(X=3)&={\frac {9}{64}}&p(X=7)&=2{\frac {3}{64}}&p(X=10)&={\frac {1}{64}}\end{aligned}}$ and thus the average $\mu$ is found to be ${\begin{aligned}\mu &=0p(X=0)+1p(X=1)+3p(X=3)+5p(X=5)+7p(X=7)+10p(X=10)\\&=2{\frac {12}{64}}+3{\frac {9}{64}}+10{\frac {4}{64}}+14{\frac {3}{64}}+10{\frac {1}{64}}\quad ={\frac {143}{64}}\approx 2.23\end{aligned}}$

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