Question

use the die and spinner to answer the questions below.
an experiment consists of rolling a die and spinning a spinner.
draw a tree diagram, make a list of outcomes and write the total number of outcomes.

1. what is the probability of getting a sum of 4? (write in lowest terms)
2. what is the probability of getting a sum of 5 or 3? (write in lowest terms)
3. what is the probability of getting a sum less than 6?
4. what is the probability of getting a sum of 8?

Explanation:

Step1: Find total possible outcomes

A die has 6 outcomes, spinner has 4 outcomes. Total outcomes: $6 \times 4 = 24$

Step2: List all outcomes

Outcomes are pairs (die roll, spinner number):
(1,1), (1,2), (1,3), (1,4),
(2,1), (2,2), (2,3), (2,4),
(3,1), (3,2), (3,3), (3,4),
(4,1), (4,2), (4,3), (4,4),
(5,1), (5,2), (5,3), (5,4),
(6,1), (6,2), (6,3), (6,4)

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For Question 1:

Step1: Identify sum=4 outcomes

Pairs where die + spinner = 4: (1,3), (2,2), (3,1) → 3 outcomes

Step2: Calculate probability

Probability = $\frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{3}{24} = \frac{1}{8}$

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For Question 2:

Step1: Identify sum=5 or 3 outcomes

Sum=3: (1,2), (2,1) → 2 outcomes
Sum=5: (1,4), (2,3), (3,2), (4,1) → 4 outcomes
Total favorable: $2 + 4 = 6$

Step2: Calculate probability

Probability = $\frac{6}{24} = \frac{1}{4}$

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For Question 3:

Step1: Identify sum < 6 outcomes

Sum=2: (1,1)
Sum=3: (1,2), (2,1)
Sum=4: (1,3), (2,2), (3,1)
Sum=5: (1,4), (2,3), (3,2), (4,1)
Total favorable: $1 + 2 + 3 + 4 = 10$

Step2: Calculate probability

Probability = $\frac{10}{24} = \frac{5}{12}$

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For Question 4:

Step1: Identify sum=8 outcomes

Pairs where die + spinner = 8: (4,4), (5,3), (6,2) → 3 outcomes

Step2: Calculate probability

Probability = $\frac{3}{24} = \frac{1}{8}$

Answer:

Total number of outcomes: 24

1. $\frac{1}{8}$
2. $\frac{1}{4}$
3. $\frac{5}{12}$
4. $\frac{1}{8}$

(Tree diagram note: First branch set is die rolls 1-6, each branching to spinner numbers 1-4, with all 24 end nodes matching the listed outcomes)