Question

1. a letter is chosen from the word rock and a standard die is rolled.

sample space:
a) $p(\text{at most } 4)$
b) $p(\text{c})$
c) $p(\text{a vowel then an even number})$
d) $p(\text{r or k then } 6)$
e) $p(\text{a consonant or an odd number})$

Explanation:

First, define the sample space:
The word ROCK has 4 distinct letters: {R, O, C, K}. A standard die has 6 outcomes: {1,2,3,4,5,6}. The total sample space size is $4 \times 6 = 24$, with each outcome being (letter, number).

Step1: Define "at most 4"

At most 4 means die rolls 1,2,3,4 (4 outcomes).
Probability formula: $P(\text{at most }4) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}}$
Expression: $\frac{4 \times 4}{24} = \frac{16}{24}$

Step2: Simplify the fraction

Reduce $\frac{16}{24}$ by GCD 8.
Expression: $\frac{16 \div 8}{24 \div 8} = \frac{2}{3}$

Step3: Calculate P(C)

Letter C is 1 of 4 letters, die has 6 outcomes.
Expression: $\frac{1 \times 6}{24} = \frac{6}{24}$

Step4: Simplify P(C)

Reduce $\frac{6}{24}$ by GCD 6.
Expression: $\frac{6 \div 6}{24 \div 6} = \frac{1}{4}$

Step5: Define vowel and even number

Vowel in ROCK is O (1 letter), even numbers are 2,4,6 (3 outcomes).
Expression: $\frac{1 \times 3}{24} = \frac{3}{24}$

Step6: Simplify the vowel-even probability

Reduce $\frac{3}{24}$ by GCD 3.
Expression: $\frac{3 \div 3}{24 \div 3} = \frac{1}{8}$

Step7: Define R/K then 6

R or K is 2 letters, die roll 6 is 1 outcome.
Expression: $\frac{2 \times 1}{24} = \frac{2}{24}$

Step8: Simplify R/K then 6 probability

Reduce $\frac{2}{24}$ by GCD 2.
Expression: $\frac{2 \div 2}{24 \div 2} = \frac{1}{12}$

Step9: Use addition rule for e)

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$
A = consonant (3 letters: R,C,K), B = odd number (3 outcomes:1,3,5)
$P(A) = \frac{3 \times 6}{24} = \frac{18}{24}$, $P(B) = \frac{4 \times 3}{24} = \frac{12}{24}$, $P(A \text{ and } B) = \frac{3 \times 3}{24} = \frac{9}{24}$
Expression: $\frac{18}{24} + \frac{12}{24} - \frac{9}{24} = \frac{21}{24}$

Step10: Simplify e) probability

Reduce $\frac{21}{24}$ by GCD 3.
Expression: $\frac{21 \div 3}{24 \div 3} = \frac{7}{8}$

Answer:

a) $\frac{2}{3}$
b) $\frac{1}{4}$
c) $\frac{1}{8}$
d) $\frac{1}{12}$
e) $\frac{7}{8}$