Question

Determine whether the events are independent. (See Examples I and 2.) You have one red apple and three green apples in a bowl. You randomly select one apple to eat now and another apple for your lunch. Use a sample space to determine whether randomly selecting a green apple first and randomly selecting a green apple second are independent events.

Short answer

No, the events are not independent.

Step-by-step solution

Defining the Events

Let event A be 'selecting a green apple first' and event B be 'selecting a green apple second'. There are 4 apples in total: one red apple and three green apples.

Calculating the Probability of Each Event

The probability of event A, \( P(A) \), is the number of green apples (3) divided by the total number of apples (4). Therefore, \( P(A) = 3/4 \). If event A has occurred, there are 2 green apples and 1 red apple left. So, the probability of event B given that event A has occurred, \( P(B|A) \), is the number of green apples left (2) divided by the total number of apples left (3). Therefore, \( P(B|A) = 2/3 \).

Determine If the Events are Independent

Events A and B are independent if \( P(B|A) = P(B) \). The probability of event B, \( P(B) \), can be calculated as the number of ways to select a green apple (3) out of total ways to select any apple (4), which gives \( P(B) = 3/4 \). However, \( P(B|A) = 2/3 \) is not equal to \( P(B) = 3/4 \), which means event B does depend on event A. Therefore, selecting a green apple first and selecting a green apple second are not independent events.

Key concepts

Understanding Probability

Probability is a measure that describes how likely an event is to occur.
It's usually expressed as a fraction or decimal ranging from 0 to 1, where 0 means the event is impossible and 1 means the event is certain.
In our apple scenario, probability helps us predict the chances of picking a green apple from the bowl.

• The probability of picking one green apple out of four apples is calculated by dividing the number of favorable outcomes (3 green apples) by the total number of possible outcomes (4 apples). This gives us the probability of selecting a green apple as \( P(A) = \frac{3}{4} \).
• Probability can be influenced by prior events, which ties into the concept of independence.

A solid grasp of probability allows us to quantify randomness and make informed predictions in various situations.

The Role of Sample Space

A sample space encompasses all possible outcomes of an experiment. In our case, it includes every conceivable combination of apples you could select.
Understanding the sample space is crucial because it sets the foundation for calculating probabilities.

• The sample space for our apple exercise involves initially four possible apples: 3 green and 1 red.
• If you pick a green apple first, the remaining sample space changes, now comprising 2 green apples and 1 red apple.
• Every time an apple is selected, the sample space and probability calculations might be affected since there are fewer apples left.

Comprehending sample spaces can help to clarify how different combinations and sequences of events affect probability.

Conditional Probability and Independence

Conditional probability investigates how the occurrence of one event affects the probability of another event.
It represents the likelihood of event B occurring given that event A has occurred, expressed as \( P(B|A) \).

• For the apple exercise, the conditional probability \( P(B|A) = \frac{2}{3} \) describes selecting a second green apple when one has already been chosen.
• If the conditional probability equals the standalone probability of an event, each is independent.
• However, here \( P(B|A) eq P(B) \), indicating dependence; the choice of the first apple influences the probability of the second.

Understanding conditional probability helps in differentiating between dependent and independent events and aids in analyzing complex probability scenarios.