Chapter 8: Problem 9
A coin that is twice as likely to show heads than it is tails is tossed three times. (a) Describe this experiment as a Bernoulli process. (b) Use a tree diagram to assign a probability density,
Short Answer
Expert verified
Experiment is a Bernoulli process with probabilities \( P(H) = \frac{2}{3} \), \( P(T) = \frac{1}{3} \); represented by a tree diagram showing all probabilities.
Step by step solution
01
Understand the Bernoulli process
A Bernoulli process is a sequence of independent experiments where each experiment results in a binary outcome, such as heads or tails. Here, we have an unfair coin, where the probability of heads is twice that of tails.
02
Define probabilities for heads and tails
Let the probability of obtaining tails be \( p \). Since the probability of heads is twice that of tails, the probability of obtaining heads is \( 2p \). The sum of probabilities for heads and tails must equal 1: \( 2p + p = 1 \). Solving this, we find \( p = \frac{1}{3} \) for tails and \( 2p = \frac{2}{3} \) for heads.
03
Describe the experiment as a Bernoulli process
The experiment involves tossing the coin three times. Each toss is an independent trial with two possible outcomes: heads and tails. The probability of heads is \( \frac{2}{3} \) and tails is \( \frac{1}{3} \). This setup matches the definition of a Bernoulli process.
04
Construct the tree diagram
A tree diagram will help visualize all possible outcomes for three coin tosses along with their probabilities. Start with the first toss node branching into two, heads \( (H) \) with \( \frac{2}{3} \) and tails \( (T) \) with \( \frac{1}{3} \). For each subsequent toss, create similar branches from each previous outcome, ensuring each branch reflects the appropriate probability.
05
Calculate probabilities for each path
Determine the probability for each sequence of outcomes from the tree diagram. Multiply along the branches. For example, the probability for the sequence HHT (head, head, tail) is \( \frac{2}{3} \times \frac{2}{3} \times \frac{1}{3} = \frac{4}{27} \). Do this for all possible sequences.
06
Verify the probabilities
Ensure the sum of probabilities for all branches is 1, verifying that we have accounted for all possible outcomes accurately in the tree diagram. This check confirms the tree diagram represents a valid probability distribution.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Probability Distribution
When dealing with a Bernoulli process involving a coin toss, especially one that is unfair like in this problem, understanding the probability distribution is key. A probability distribution gives us the probabilities of all possible outcomes. Here, since the coin is twice as likely to land on heads as tails, the probabilities need careful consideration.
First, let us denote the probability of the coin landing on tails as \( p \). If heads are twice as likely as tails, it follows that the probability of getting heads is \( 2p \). As the sum of all probabilities must equal 1, we set up the equation \( 2p + p = 1 \). Solving gives \( p = \frac{1}{3} \) for tails and \( 2p = \frac{2}{3} \) for heads.
The probability distribution for three coin tosses includes all combinations and their associated probabilities. Using these probabilities, we calculate the likelihood of each potential outcome. This involves listing all sequences, such as HHH, HHT, and so on, and determining the probability for each sequence by multiplying along the path in the tree diagram.
First, let us denote the probability of the coin landing on tails as \( p \). If heads are twice as likely as tails, it follows that the probability of getting heads is \( 2p \). As the sum of all probabilities must equal 1, we set up the equation \( 2p + p = 1 \). Solving gives \( p = \frac{1}{3} \) for tails and \( 2p = \frac{2}{3} \) for heads.
The probability distribution for three coin tosses includes all combinations and their associated probabilities. Using these probabilities, we calculate the likelihood of each potential outcome. This involves listing all sequences, such as HHH, HHT, and so on, and determining the probability for each sequence by multiplying along the path in the tree diagram.
Independent Trials
In probability theory, an independent trial is an experiment where the outcome of one event does not affect the outcome of another. This is a core component of a Bernoulli process. Each coin toss in our exercise is an independent trial with two potential outcomes: heads or tails.
The independence of trials is crucial because it implies that the probability of heads or tails remains constant across each toss. For example, even after getting heads twice in a row, the probability of heads on the next toss remains \( \frac{2}{3} \), and for tails, \( \frac{1}{3} \).
This concept allows us to calculate the probability of a sequence of outcomes by multiplying the probabilities of individual outcomes. Since tossing the coin three times involves three independent trials, we can confidently calculate the probability of any sequence, such as HHT or TTH, without considering other sequences.
The independence of trials is crucial because it implies that the probability of heads or tails remains constant across each toss. For example, even after getting heads twice in a row, the probability of heads on the next toss remains \( \frac{2}{3} \), and for tails, \( \frac{1}{3} \).
This concept allows us to calculate the probability of a sequence of outcomes by multiplying the probabilities of individual outcomes. Since tossing the coin three times involves three independent trials, we can confidently calculate the probability of any sequence, such as HHT or TTH, without considering other sequences.
Tree Diagram
A tree diagram is a powerful visual tool in probability that helps organize and calculate the potential outcomes and their probabilities for a series of events, particularly useful here with our unfair coin.
To start building a tree diagram for our exercise, begin by representing the first coin toss with two branches leading to the two possible outcomes: heads (H) with a probability of \( \frac{2}{3} \) and tails (T) with \( \frac{1}{3} \).
For the second toss, each outcome from the first toss splits again into two branches for the subsequent head or tail, keeping their respective probabilities. Continue this process for the third toss, resulting in a full tree diagram with all possible sequences of outcomes: HHH, HHT, HTT, etc.
The tree diagram not only helps to visualize the different paths but also aids in calculating the probabilities for each sequence. For instance, the probability of the sequence HHT (head-head-tail) involves multiplying probabilities along that specific path: \( \frac{2}{3} \times \frac{2}{3} \times \frac{1}{3} = \frac{4}{27} \).
To start building a tree diagram for our exercise, begin by representing the first coin toss with two branches leading to the two possible outcomes: heads (H) with a probability of \( \frac{2}{3} \) and tails (T) with \( \frac{1}{3} \).
For the second toss, each outcome from the first toss splits again into two branches for the subsequent head or tail, keeping their respective probabilities. Continue this process for the third toss, resulting in a full tree diagram with all possible sequences of outcomes: HHH, HHT, HTT, etc.
The tree diagram not only helps to visualize the different paths but also aids in calculating the probabilities for each sequence. For instance, the probability of the sequence HHT (head-head-tail) involves multiplying probabilities along that specific path: \( \frac{2}{3} \times \frac{2}{3} \times \frac{1}{3} = \frac{4}{27} \).
Unfair Coin
An unfair coin is one where the probabilities of landing on heads or tails are not equal. This concept is central to our exercise, where the coin is twice as likely to land heads as tails.
Understanding how an unfair coin works is essential for setting up correct probability distributions and analyzing outcomes. In this scenario, fairness is skewed deliberately, which directly influences the calculations as seen in our solutions.
This imbalance affects every probability calculation, from the initial setting of heads at \( \frac{2}{3} \) and tails at \( \frac{1}{3} \), to the resulting probabilities of each sequence in a three-toss scenario. When working with an unfair coin, maintained awareness of these altered probabilities ensures accuracy and helps avoid incorrect assumptions that might arise if the coin were presumed fair.
Understanding how an unfair coin works is essential for setting up correct probability distributions and analyzing outcomes. In this scenario, fairness is skewed deliberately, which directly influences the calculations as seen in our solutions.
This imbalance affects every probability calculation, from the initial setting of heads at \( \frac{2}{3} \) and tails at \( \frac{1}{3} \), to the resulting probabilities of each sequence in a three-toss scenario. When working with an unfair coin, maintained awareness of these altered probabilities ensures accuracy and helps avoid incorrect assumptions that might arise if the coin were presumed fair.
