Question

Tell whether the events are independent or dependent. Explain your reasoning. A box of granola bars contains an assortment of flavors. You randomly choose a granola bar and eat it. Then you randomly choose another bar. Event \(\boldsymbol{A}\) : You choose a coconut almond bar first. Event \(\boldsymbol{B}\) : You choose a cranberry almond bar second.

Short answer

The events are independent because choosing a coconut almond bar first does not change the probability of picking a cranberry almond bar second.

Step-by-step solution

Understand the Situation

You are choosing two granola bars in sequence, without replacement. This means that after you choose the first one, the total number of bars goes down by 1.

Identify the Events

Event A: You choose a coconut almond bar first. Event B: You choose a cranberry almond bar second.

Determine Dependence or Independence

The choice of a coconut almond bar first (Event A) does not affect the likelihood of choosing a cranberry almond bar second (Event B). The flavors of the bars are distinct, and choosing a coconut almond bar first does not remove any cranberry almond bars from the box. Therefore, it does not change the probability of getting a cranberry almond bar in the second pick.

Final Answer

Based on the reasoning above, Event A and Event B are considered independent because the outcome of Event A does not change the probability of Event B.

Key concepts

Probability

Probability is the measure of how likely an event is to occur. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, if you have a box with 10 granola bars of different flavors, and you want to pick a coconut almond bar, the probability is the number of coconut almond bars divided by 10. This gives the likelihood of that specific event happening. Probability is usually expressed as a number between 0 and 1. A probability of 0 means the event cannot happen, while a probability of 1 means the event is certain to happen.

Dependent Events

When we talk about dependent events, we refer to events where the outcome of one event affects the probability of the other event. If Event A affects Event B, they are dependent. In the exercise, you pick two granola bars in sequence. Normally, this situation suggests dependency because one pick affects the total left, changing the odds of picking specific flavors. However, in this case, the choice of a coconut almond bar doesn't influence picking a cranberry almond bar because their presence in the box is independent of each other. This makes Events A and B independent, since picking one flavor doesn't remove another from the box.

Combinatorics

Combinatorics is a branch of mathematics centered around counting, arrangement, and combination of objects. It plays a significant role in probability because it helps determine the total number of ways events can happen. In scenarios like choosing granola bars, you often utilize combinatorics to find the total possible outcomes and the favorable outcomes. For example, if there are multiple coconut and cranberry almond bars, combinatorics can help calculate how many combinations consist of selecting these specific flavors in sequence. Combinatorics includes important principles like permutations, which count arrangements where order matters, and combinations, focusing on selections where order doesn't matter. This is essential for situations involving multiple steps, like picking granola bars one after the other.