Question

1. a jar contains 32 red marbles and 28 blue marbles. what is the probability that a randomly selected marble is red?
2. a letter in the word restoration is randomly selected. what is the probability of selecting a vowel?
3. two dice are rolled. what is the probability that the sum of the two dice is at least 9 on the next roll?
4. what is the probability of drawing a heart or a club from a standard deck of cards?

Explanation:

Step1: Calculate total marbles

Total marbles = $32 + 28 = 60$

Step2: Find red marble probability

Probability = $\frac{\text{Number of red marbles}}{\text{Total marbles}} = \frac{32}{60} = \frac{8}{15}$

Step1: Count letters and vowels

Word: RESTORATION. Total letters = 11. Vowels (E, O, A, I, O) = 5

Step2: Find vowel probability

Probability = $\frac{\text{Number of vowels}}{\text{Total letters}} = \frac{5}{11}$

Step1: List valid dice sums

Sums at least 9: (3,6), (4,5), (4,6), (5,4), (5,5), (5,6), (6,3), (6,4), (6,5), (6,6). Total valid outcomes = 10
Total possible dice outcomes = $6 \times 6 = 36$

Step2: Find sum probability

Probability = $\frac{\text{Valid outcomes}}{\text{Total outcomes}} = \frac{10}{36} = \frac{5}{18}$

Step1: Count hearts, clubs, total cards

Hearts = 13, Clubs = 13, Total cards = 52

Step2: Find combined probability

Probability = $\frac{\text{Hearts} + \text{Clubs}}{\text{Total cards}} = \frac{13+13}{52} = \frac{26}{52} = \frac{1}{2}$

Answer:

1. $\frac{8}{15}$
2. $\frac{5}{11}$
3. $\frac{5}{18}$
4. $\frac{1}{2}$