Question

there are five solids called platonic solids. each platonic solid has congruent faces and congruent angles, and they are shown below. use the figures to compute the probabilities. express your answers as fractions.
a. an octahedron has 8 faces. if the octahedron faces are labeled with the numbers 1 - 8, and it is rolled, find the probability that it will land on a prime number.
b. the icosahedron has 20 congruent triangular faces. if they are labeled 1 - 20, and the icosahedron is rolled, what is the probability that it lands on a multiple of 3?
c. the dodecahedron has 12 congruent pentagonal faces. if they are labeled 1 - 12, and the dodecahedron is rolled, find the probability that it lands on a multiple of 3 and 2.
d. a tetrahedron has four congruent triangular faces labeled 10, 20, 30, and 40. let t represent \the tetrahedron was rolled.\ let s represent the event \lands on a multiple of 7.\ find p(s|t).

Explanation:

Step1: Identify prime numbers from 1 - 8

Prime numbers from 1 - 8 are 2, 3, 5, 7. There are 4 prime numbers out of 8.

Step2: Calculate probability

The probability formula is $P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. So for an octahedron with 8 faces labeled 1 - 8, the probability of landing on a prime number is $\frac{4}{8}=\frac{1}{2}$.

Step3: Identify multiples of 3 from 1 - 20

Multiples of 3 from 1 - 20 are 3, 6, 9, 12, 15, 18. There are 6 multiples of 3 out of 20. So for an icosahedron with 20 faces labeled 1 - 20, the probability of landing on a multiple of 3 is $\frac{6}{20}=\frac{3}{10}$.

Step4: Identify multiples of 3 and 2 (i.e., multiples of 6) from 1 - 12

Multiples of 6 from 1 - 12 are 6, 12. There are 2 multiples of 6 out of 12. So for a dodecahedron with 12 faces labeled 1 - 12, the probability of landing on a multiple of 3 and 2 is $\frac{2}{12}=\frac{1}{6}$.

Step5: Identify multiples of 7 from 10, 20, 30, 40

None of 10, 20, 30, 40 is a multiple of 7. So the number of favorable outcomes is 0. For a tetrahedron with 4 faces labeled 10, 20, 30, 40, the probability of landing on a multiple of 7 is $\frac{0}{4}=0$.

Answer:

a. $\frac{1}{2}$
b. $\frac{3}{10}$
c. $\frac{1}{6}$
d. $0$