Question

independent events
independent practice
use your understanding of probability and independent events to answer the questions below.

1. roberto will flip a coin and spin the spinner shown.

a. make a tree diagram or list to represent the sample space
b. what is the probability the coin will land on tails and the spinner will land on a 4?
c. what is the probability the coin will land on heads and the spinner will land on an odd number?

Explanation:

Part a:

Step1: Identify outcomes for coin

A coin has 2 outcomes: Heads (H) and Tails (T).

Step2: Identify outcomes for spinner

The spinner has numbers 1,2,3,4,5,6,7 (7 sections? Wait, looking at the spinner: numbers 1,2,3,4,5,6,7? Wait the spinner in the image: let's count the sections. The spinner is divided into 7? Wait no, the numbers are 1,2,3,4,5,6,7? Wait the user's image: the spinner has 1,2,3,4,5,6,7? Wait no, maybe 7? Wait no, the numbers are 1,2,3,4,5,6,7? Wait the spinner is labeled 1,2,3,4,5,6,7? Wait maybe 7 sections. Wait no, maybe 7? Wait the coin has 2 outcomes, spinner has 7? Wait no, looking at the spinner: 1,2,3,4,5,6,7 – 7 numbers. Wait but maybe I miscounted. Wait the spinner in the image: let's see, the numbers are 1,2,3,4,5,6,7 – 7 sections. So spinner outcomes: 1,2,3,4,5,6,7.

Step3: List sample space

For each coin outcome (H, T), pair with each spinner outcome (1,2,3,4,5,6,7). So the sample space is:

• H with 1, H with 2, H with 3, H with 4, H with 5, H with 6, H with 7
• T with 1, T with 2, T with 3, T with 4, T with 5, T with 6, T with 7

Alternatively, as a list: {(H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (H,7), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6), (T,7)}

Step1: Probability of coin landing on Tails

A coin has 2 outcomes (H, T), so \(P(\text{Tails}) = \frac{1}{2}\).

Step2: Probability of spinner landing on 4

Spinner has 7 outcomes (1 - 7), so \(P(\text{spinner on 4}) = \frac{1}{7}\).

Step3: Multiply probabilities (independent events)

Since coin flip and spinner spin are independent, \(P(\text{Tails and 4}) = P(\text{Tails}) \times P(\text{spinner on 4}) = \frac{1}{2} \times \frac{1}{7} = \frac{1}{14}\).

Step1: Probability of coin landing on Heads

\(P(\text{Heads}) = \frac{1}{2}\).

Step2: Identify odd numbers on spinner

Odd numbers: 1, 3, 5, 7 (4 outcomes). Total spinner outcomes: 7. So \(P(\text{spinner on odd}) = \frac{4}{7}\).

Step3: Multiply probabilities (independent events)

\(P(\text{Heads and odd}) = P(\text{Heads}) \times P(\text{spinner on odd}) = \frac{1}{2} \times \frac{4}{7} = \frac{4}{14} = \frac{2}{7}\).

Answer:

(for part a):
The sample space can be represented as a list: \(\{(H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (H,7), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6), (T,7)\}\) (or using a tree diagram with coin branches (H, T) and each branching into spinner numbers 1 - 7).

Part b: