Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 3
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
Expert verified
The events are independent.
Step by step solution
01
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.
02
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.
03
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.
04
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 \( \{HHT, HTH, THH, HTT, THT, TTH\} \).
05
Count the outcomes where not all flips are identical
There are 6 outcomes: \( HHT, HTH, THH, HTT, THT, \) and \( TTH \).
06
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.
07
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.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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).
- 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.
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.
- 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.
