Problem 16
Question
Three fair coins are tossed. If both heads and tails appears, then the probability that exactly one head appears is (a) \(3 / 8\) (b) \(1 / 6\) (c) \(1 / 2\) (d) \(1 / 3\)
Step-by-Step Solution
Verified Answer
The probability is \(\frac{1}{2}\).
1Step 1: Identify Possible Outcomes
When three fair coins are tossed, each coin has two possible outcomes: Heads (H) or Tails (T). Therefore, the total number of outcomes is given by \(2^3 = 8\). The possible outcomes are: HHH, HHT, HTH, THH, TTH, THT, HTT, TTT.
2Step 2: Determine Outcomes with Both Heads and Tails
Outcomes where both heads and tails appear are those that are not all heads (HHH) or all tails (TTT). These outcomes are HHT, HTH, THH, TTH, THT, and HTT, making a total of 6 outcomes.
3Step 3: Identify Outcomes with Exactly One Head
Out of the 8 possible outcomes, those with exactly one head are HTT, THT, and TTH. This gives us a total of 3 outcomes with exactly one head.
4Step 4: Calculate Probability of Exactly One Head Given Both Heads and Tails
Given that both heads and tails appear, there are 6 such outcomes (as identified in Step 2). Out of these 6 outcomes, 3 have exactly one head (as identified in Step 3). Therefore, the probability is \(\frac{3}{6} = \frac{1}{2}\).
Key Concepts
Coin TossSample SpaceConditional ProbabilityProbability Theory
Coin Toss
A coin toss is a simple yet fundamental event in probability theory. Every coin toss involves an unbiased coin with two possible outcomes — heads (H) or tails (T). When tossing multiple coins, each coin acts independently of the others. Thus, the outcomes do not influence each other.
For instance, when you toss three coins, as in our exercise, each coin contributes independently to the overall set of outcomes. This independence is crucial because it allows us to determine the joint outcomes by considering every coin's possibilities.
This makes a coin toss an ideal illustration of basic probability principles. It demonstrates how random experiments work and how outcomes are calculated when combining multiple independent random events.
For instance, when you toss three coins, as in our exercise, each coin contributes independently to the overall set of outcomes. This independence is crucial because it allows us to determine the joint outcomes by considering every coin's possibilities.
This makes a coin toss an ideal illustration of basic probability principles. It demonstrates how random experiments work and how outcomes are calculated when combining multiple independent random events.
Sample Space
The sample space is a comprehensive list of all possible results of a random experiment. For a coin toss, this means accounting for every variation of heads and tails.
When three coins are tossed, as in the exercise, the sample space consists of all possible sequences of their outcomes. This is calculated as follows: each coin has 2 possible results – H or T. So, calculating for three coins, we have\[2^3 = 8\]possible outcomes, namely: HHH, HHT, HTH, THH, TTH, THT, HTT, and TTT.
The sample space is vital because it sets the stage for identifying specific events and further calculating probabilities. Without knowing the entirety of possible outcomes, determining the probability of any particular event would be incomplete.
When three coins are tossed, as in the exercise, the sample space consists of all possible sequences of their outcomes. This is calculated as follows: each coin has 2 possible results – H or T. So, calculating for three coins, we have\[2^3 = 8\]possible outcomes, namely: HHH, HHT, HTH, THH, TTH, THT, HTT, and TTT.
The sample space is vital because it sets the stage for identifying specific events and further calculating probabilities. Without knowing the entirety of possible outcomes, determining the probability of any particular event would be incomplete.
Conditional Probability
Conditional probability is the likelihood of an event occurring, given that another event has already happened. It's a refinement of simple probability that limits the sample space.
In the exercise at hand, you focus only on outcomes where both heads and tails appear. This new context restricts the number of outcomes to those that include at least one head and one tail, thus filtering out HHH and TTT.
With the revised outcomes (HHT, HTH, THH, TTH, THT, HTT) numbering 6, the conditional probability of exactly one head out of these is determined by counting the relevant outcomes — HTT, THT, TTH — and then dividing by the total conditional outcomes:\[P(\text{Exactly one head | Both heads and tails appear}) = \frac{3}{6} = \frac{1}{2}\]Understanding conditional probability is essential, especially as it applies to real-world situations where conditions often limit the range of potential outcomes.
In the exercise at hand, you focus only on outcomes where both heads and tails appear. This new context restricts the number of outcomes to those that include at least one head and one tail, thus filtering out HHH and TTT.
With the revised outcomes (HHT, HTH, THH, TTH, THT, HTT) numbering 6, the conditional probability of exactly one head out of these is determined by counting the relevant outcomes — HTT, THT, TTH — and then dividing by the total conditional outcomes:\[P(\text{Exactly one head | Both heads and tails appear}) = \frac{3}{6} = \frac{1}{2}\]Understanding conditional probability is essential, especially as it applies to real-world situations where conditions often limit the range of potential outcomes.
Probability Theory
Probability theory is a branch of mathematics that deals with quantifying the likelihood of different outcomes in random events. It's a fundamental theory used across various fields like statistics, finance, science, and engineering.
This exercise showcases essential probability concepts: events, outcomes, and conditional probabilities. Probability theory helps us understand the mechanics of events like coin tosses, predicting how often certain outcomes will occur in the long run.
This exercise showcases essential probability concepts: events, outcomes, and conditional probabilities. Probability theory helps us understand the mechanics of events like coin tosses, predicting how often certain outcomes will occur in the long run.
- Events: These are the results or occurrences that we observe in a given experiment.
- Outcomes: These are the possible end results of an experiment. Each unique sequence in the coin toss is an outcome.
- Probabilities: These express the chances of particular outcomes or events, grounded in the complete understanding of possible outcomes or sample spaces.
Other exercises in this chapter
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