Question

when you are done, check the solutions with a math teacher or tutor. tell whether the events are independent or dependent. explain your reasoning. 5. your mp3 player contains hip - hop and rock songs. you randomly choose a song. then you randomly choose another song without repeating song choices. event a: you choose a hip - hop song first. event b: you choose a rock song second. determine whether the events are independent. (see example 1 & 2) 10. a vase contains four white roses and one red rose. you randomly select two roses to take home. use a sample space to determine whether randomly selecting a white rose first and randomly selecting a white rose second are independent events.

Explanation:

Problem 5

Step1: Recall Independent/Dependent Events

Independent events: \( P(A \cap B) = P(A) \times P(B) \), or the outcome of one doesn't affect the other. Dependent: outcome of one affects the other (e.g., without replacement).

Step2: Analyze Event A and B

Event A: Choose hip - hop first (without replacement). After choosing a hip - hop song, the total number of songs decreases by 1, and the number of rock songs remains the same, but the total number of songs for the second choice is less. So the probability of Event B (choosing rock second) depends on whether Event A occurred (since we don't replace the first song).

Step1: Define Events and Sample Space

Let \( W_1, W_2, W_3, W_4 \) be white roses and \( R \) be the red rose. The sample space for selecting two roses (without replacement) is: \(\{(W_1,W_2),(W_1,W_3),(W_1,W_4),(W_1,R),(W_2,W_1),(W_2,W_3),(W_2,W_4),(W_2,R),(W_3,W_1),(W_3,W_2),(W_3,W_4),(W_3,R),(W_4,W_1),(W_4,W_2),(W_4,W_3),(W_4,R),(R,W_1),(R,W_2),(R,W_3),(R,W_4)\}\)
Let Event A: Select a white rose first. Event B: Select a white rose second.

Step2: Calculate \( P(A) \)

Total number of ways to select the first rose: 5 (4 white + 1 red). Number of white roses: 4. So \( P(A)=\frac{4}{5} \)

Step3: Calculate \( P(B|A) \) (probability of B given A)

If we selected a white rose first, there are 3 white roses and 1 red rose left (total 4 roses). So \( P(B|A)=\frac{3}{4} \)

Step4: Calculate \( P(B) \)

To calculate \( P(B) \), we consider two cases: first rose is white or first rose is red.
Case 1: First rose is white (probability \(\frac{4}{5}\)), then second rose white: \(\frac{3}{4}\). Case 2: First rose is red (probability \(\frac{1}{5}\)), then second rose white: \(\frac{4}{4} = 1\)
Using the law of total probability: \( P(B)=\frac{4}{5}\times\frac{3}{4}+\frac{1}{5}\times1=\frac{3}{5}+\frac{1}{5}=\frac{4}{5} \)

Step5: Check Independence

For independent events, \( P(B|A)=P(B) \). But \( P(B|A)=\frac{3}{4} \) and \( P(B)=\frac{4}{5} \), and \(\frac{3}{4}
eq\frac{4}{5}\). Also, \( P(A\cap B)=\frac{4}{5}\times\frac{3}{4}=\frac{3}{5} \), and \( P(A)\times P(B)=\frac{4}{5}\times\frac{4}{5}=\frac{16}{25} \), and \(\frac{3}{5}
eq\frac{16}{25}\)

Answer:

The events are dependent. Because we do not repeat song choices (without replacement), the outcome of Event A (choosing a hip - hop song first) affects the total number of songs available for Event B (choosing a rock song second), thus changing the probability of Event B.

Problem 10