Question

In three flips of a coin, is the event of getting at most one tail independent of the event that not all flips are identical?

Short answer

The events are independent.

Step-by-step solution

Define the sample space

The sample space when flipping a coin three times is \( \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\} \). There are 8 possible outcomes.

Identify the event of getting at most one tail

For the event of getting at most one tail, the outcomes are \( \{HHH, HHT, HTH, THH\} \). This is because there are either zero or one tail.

Count the outcomes for at most one tail

There are 4 outcomes: \( HHH, HHT, HTH, \) and \( THH \) in the event of getting at most one tail.

Identify the event of not all flips being identical

The event that not all flips are identical includes all outcomes except \( HHH \) and \( TTT \). So, the outcomes are \( \{HHT, HTH, THH, HTT, THT, TTH\} \).

Count the outcomes where not all flips are identical

There are 6 outcomes: \( HHT, HTH, THH, HTT, THT, \) and \( TTH \).

Determine the intersection of the two events

The common outcomes between the two events are \( \{HHT, HTH, THH\} \), which is the intersection of the two sets.

Check for independence

Events are independent if the probability of their intersection equals the product of their probabilities. Calculate:- Probability of at most one tail: \( \frac{4}{8} = \frac{1}{2} \).- Probability of not all flips identical: \( \frac{6}{8} = \frac{3}{4} \).- Probability of intersection: \( \frac{3}{8} \).Check independence: \( \frac{1}{2} \times \frac{3}{4} = \frac{3}{8} \), which matches the intersection probability.

Key concepts

Independent Events

In probability theory, understanding independent events is crucial. Events are considered independent when the outcome of one event does not influence or change the probability of the other event. For two events, say A and B, to be independent, the probability of both events occurring (their intersection) must equal the product of their probabilities:

• Mathematically, this condition is expressed as: \[ P(A \cap B) = P(A) \times P(B) \].
• If this equation holds true, the two events do not affect each other, and they are independent.
• If not, they are dependent, meaning the outcome of one event influences the other.

Whenever dealing with independent events, it's important to verify with probabilities and their intersection.

Coin Flipping

Coin flipping is a classic example used in probability theory to demonstrate simple random events. Each flip of a coin is considered a fair event:

• You have two possible outcomes: heads (H) or tails (T).
• Each outcome has an equal chance of occurring, typically represented as a probability of \( \frac{1}{2} \) for heads and \( \frac{1}{2} \) for tails.

When a coin is flipped multiple times, these flips are considered independent events, assuming the coin is fair and conditions remain constant. This means the result of one flip does not change the probability of outcomes in subsequent flips. Whether you are flipping a coin once or multiple times, each flip offers a simple and clear example of random events in probability.

Sample Space

In probability, the sample space of an experiment is the set of all possible outcomes. For a single flip of a coin, the sample space is fairly small:

• It's \( \{H, T\} \) (Heads or Tails).

For more complex scenarios, like flipping a coin more times, the sample space grows. Consider flipping a coin three times:

• The sample space becomes \( \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\} \).
• These eight outcomes account for all the sequences of heads and tails that could theoretically occur over three flips.

Sample spaces are essential in probability theory because they give us the complete picture of all potential results, allowing us to calculate the probabilities of various events.

Intersection of Events

The intersection of events in probability refers to outcomes that are common to two or more events. When considering two sets, their intersection contains all outcomes that both sets share. This is denoted as:

• \[ A \cap B \] , where A and B are events.
• The intersection is essentially the "overlap" of these events.

Finding the intersection is vital when determining if events are independent. For example, consider two events in a three-flip sequence: >\( \text{At most one tail}\)and \(\text{Not all flips are identical} \). Their intersection would be sequences where both conditions are met, like \( \{HHT, HTH, THH\} \).

• The probability of their intersection can be calculated by taking the number of favorable outcomes in the intersection and dividing it by the total number of possible outcomes in the sample space.

Understanding and calculating intersections allow us to delve deeper into event dependencies and overall probability outcomes.