Question

Determine whether the events are independent. (See Examples I and 2.) 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.

Short answer

The product of the individual probabilities (\(\frac{4}{5} * \frac{3}{4} = \frac{12}{20} = \frac{6}{10} = 0.6\)) is equal to the joint probability (\(\frac{6}{10} = 0.6\)), which means that the events are indeed independent.

Step-by-step solution

Calculate initial probabilities

Firstly, calculate the probability of picking a white rose at first attempt. Since there are 4 white roses out of 5 roses total, the probability is \(\frac{4}{5}\). Calculate next the probability of picking a white rose at the second attempt if the first one was white. This time, we have 3 white roses and 4 total roses, so the probability is \(\frac{3}{4}\).

Calculate joint probability

Now, calculate the joint probability of both events occurring, which is first picking a white rose and then picking another white rose. Both of these events happening at the same time would initially mean we have 4 white roses and we need to choose two. The probability of that in principle is via combination: \(\frac{\binom{4}{2}}{\binom{5}{2}} = \frac{6}{10}\).

Compare probabilities

Let's compare the product of the individual probabilities from step 1 (\(\frac{4}{5} * \frac{3}{4}\)) with the joint probability from step 2 (\(\frac{6}{10}\)). If the two are equal, then the events are independent.

Key concepts

Probability Calculation

When it comes to understanding probability, it's important to start with the basics. Probability calculation is the process of determining the likelihood of an event occurring. The probability of an event is a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

To calculate the probability of a single event, you divide the number of favorable outcomes by the total number of possible outcomes in the sample space. In our vase example, the initial probability of drawing a white rose is determined by dividing the number of white roses by the total number of roses, resulting in a probability of \( \frac{4}{5} \).

It's also essential to understand that probabilities can change after the occurrence of an event, like drawing one rose out of the vase, which alters the sample space for the subsequent draw. This change is part of what we consider when determining event independence.

Sample Space

The sample space is the set of all possible outcomes in a probability experiment. It's the foundation of any probability calculation because it provides the context for determining probabilities of different events.

For our vase with five roses, when you're drawing one rose, the sample space consists of five outcomes, one for each rose. However, once a rose has been drawn, the sample space for the next draw shrinks. If you selected a white rose first, you are left with four possible outcomes for the second draw. This is a critical concept because the size and composition of the sample space affects the probability of subsequent events. Developing a clear understanding of sample space helps students maintain accuracy in probability calculations for successive events.

Joint Probability

Joint probability refers to the likelihood of two independent events happening at the same time. Calculating joint probability typically involves multiplying the probabilities of the individual events, if they are indeed independent.

In the vase example, the joint probability of picking a white rose first and a white rose second involves determining the probability of both these events occurring together. As we saw in the solution, the calculation for joint probability made use of combinations to reflect the probabilities of two successive choices from the vase.

However, it's crucial to recognize that if the outcome of one event affects the other, the events are not independent, and we cannot simply multiply the individual probabilities. Therefore, assessing independence is a key step before using joint probabilities.

Combinations in Probability

Combinations are a way to calculate the probability of selecting a group of items from a larger set, where the order of the items doesn't matter. In probability, this concept is frequently used to simplify the calculation of complex events.

When we worked out the joint probability of choosing two white roses from the vase, we used combinations to express the possible selections. We calculated \( \binom{4}{2} \) to represent the ways two white roses could be chosen from the four available, and \( \binom{5}{2} \) for the total ways to choose any two roses from the five.

Understanding how to calculate combinations is essential for problems where you're dealing with groups of items or multiple events occurring. It's a powerful tool that simplifies the analysis of probabilities in more complex sample spaces.