Four-Color Pyramid Dice Model

GeoGebra Commands & Parameters

1. LaTeX("\text{State the probability of any of the 4 faces landing on the bottom.}") → step2
2. LaTeX("\text{Each face has a 1 in 4, or 0.25, chance of landing on the bottom.}") → step2_1
3. LaTeX("\text{Thus, the probability of the green face landing on the bottom is 0.25.}") → step2_2
4. LaTeX("\text{This answer should add up to 1.}") → step3_1
5. LaTeX("\text{Write the function for the green face landing on the bottom n times in a row.}") → step4
6. LaTeX("f(n)=a^n") → step4_2
7. LaTeX("f(n) = (4^{–1})^n") → step4_4
8. LaTeX("f(n) = 4^{–n}") → step4_5
9. LaTeX("\text{Write the inverse of the function determined in Step 4.}") → step5
10. LaTeX(" \text{This will give the logarithm requested in the problem.}") → step5_1
11. LaTeX("\text{Explain what the inverse function describes in the context of this problem.}") → step6
12. LaTeX("\text{Switch the variables and change } f(n) \text{ to } f^{-1}(n):") → step5_3
13. LaTeX("f^{-1}(n)=\log_4 \, n = \log_4 \, n^{-1}") → step5_{32}
14. LaTeX("\text{This is the first step in defining the probability event. }\cr\text{There are 4 faces on the token.}") → step1_1
15. LaTeX("\text{State the number of faces on the token.}") → step1
16. LaTeX("\text{Check your answer to step 2 by calculating the chance that any}\cr\text{of the 4 faces lands on the bottom.}") → step3
17. LaTeX("\text{The probability of a blue, green, red, or white face landing on the bottom}\cr\text{is given by }0.25 + 0.25 + 0.25 + 0.25\text{, which equals 1.}") → step3_2
18. LaTeX("\text{Therefore, the inverse function is } f^{-1}(n) = \log_4 \, n^{-1}.") → step5_4
19. LaTeX(" \text{Suppose n = 4. The value of the function at n = 4 is given by the ordered}\cr\text{pair }(4,\ 4^{–4})\text{, so the corresponding point for the inverse is }(4^{-4},\ 4).\cr\text{The inverse implies that when a probability of }4^{–4}\text{ occurs, the token has }\cr\text{been rolled 4 times with the same outcome }(4^{–1})\text{ each time.}") → step6_1
20. LaTeX("\text{This generalizes the result of step 2 for any number of rolls of the token}\cr\text{Let }a = \frac{1}{4} \text{ in the function }f(n) = a^n \text{ and simplify.}") → step4_1
21. LaTeX("f(n)=\bigg(\frac{1}{4}\bigg)^n") → step4_3
22. LaTeX((((((("\text{The inverse of } f(n) = " * a) * "^{-n} \text { is } \log_") * a) * "\,f(n) = -n, \text{ which simplies to } \cr\ n=-\log_") * a) * " \, f(n).")) → step5_2
23. LaTeX("\text{Problem: }\cr \text{A pyramid-shaped token for a board game consists of 4 congruent equilateral faces. Each face}\cr\text{is a different color: blue, green, red, or white. Each face is equally likely to end up on the}\cr\text{bottom if the token is rolled on the game board. What is the probability of the green face}\cr\text{landing on the bottom? The probability of the same event a occurring n times is given by the}\cr \text{function } f(n) = a^n \text{. Write a function for the green face landing on the bottom n times in a}\cr\text{row. Then, write the inverse of the function, and explain what the inverse function describes}\cr\text{in the context of this problem.}") → Problem