Question

A coin that is twice as likely to show heads than it is tails is tossed three times, Suppose we are only interested in the number of heads. (a) Formulate this experiment in terms of a sample space with outcomes that give the number of heads. (b) Assign a probability density function to this sample space. (c) Describe the event "at least two heads" as a set of outcomes in this sample space. (d) What is the probability of the event in part (c)?

Short answer

The probability of at least two heads is \(\frac{20}{27}\).

Step-by-step solution

Understanding the Experiment

This experiment involves tossing a biased coin three times where heads is twice as likely as tails. We are interested in the number of heads that appear in the three tosses.

Formulating the Sample Space

The sample space, based on the number of heads in three tosses, can be given as \(S = \{0, 1, 2, 3\}\). These represent 0, 1, 2, or 3 heads showing up in the three tosses.

Defining the Probability of Each Toss

Since heads is twice as likely as tails, let the probability of tails be \(P(T) = p\). Thus, \(P(H) = 2p\). Since the total probability must equal 1, we have \(p + 2p = 1\), which implies \(p = \frac{1}{3}\). Consequently, \(P(H) = \frac{2}{3}\).

Determining the Sample Space Probability Density Function

Using the binomial probability formula for each outcome \(k\), where the number of trials \(n = 3\), we have the probability density function \(P(X = k) = \binom{3}{k} \left(\frac{2}{3}\right)^k \left(\frac{1}{3}\right)^{3-k}\). Calculate this for \(k = 0, 1, 2, 3\).

Calculating Probabilities for Each Outcome

- \(P(0) = \binom{3}{0} \left(\frac{2}{3}\right)^0 \left(\frac{1}{3}\right)^3 = \frac{1}{27}\).- \(P(1) = \binom{3}{1} \left(\frac{2}{3}\right)^1 \left(\frac{1}{3}\right)^2 = \frac{6}{27} = \frac{2}{9}\).- \(P(2) = \binom{3}{2} \left(\frac{2}{3}\right)^2 \left(\frac{1}{3}\right)^1 = \frac{12}{27} = \frac{4}{9}\).- \(P(3) = \binom{3}{3} \left(\frac{2}{3}\right)^3 \left(\frac{1}{3}\right)^0 = \frac{8}{27}\).

Describing the Event 'At Least Two Heads'

The event of 'at least two heads' in the sample space \(S = \{0, 1, 2, 3\}\) is described by the set \(\{2, 3\}\).

Calculating the Probability of the Event 'At Least Two Heads'

To find this probability, sum the probabilities of obtaining 2 and 3 heads: \(P(2) + P(3) = \frac{4}{9} + \frac{8}{27}\). Converting \(\frac{4}{9}\) to \(\frac{12}{27}\) and adding gives \(\frac{12}{27} + \frac{8}{27} = \frac{20}{27}\).

Key concepts

Sample Space

In probability and statistics, a 'Sample Space' is a fundamental concept that refers to the set of all possible outcomes of an experiment. When you're dealing with a random process, understanding the sample space is crucial, as it represents the universe of possible results. In our example of flipping a biased coin three times, where heads are twice as likely as tails, the sample space encompasses the different counts of heads that may appear.

• The Sample Space, denoted as \( S \), includes \( \{ 0, 1, 2, 3 \} \).
• Each value represents the number of heads shown in three coin tosses: 0 heads, 1 head, 2 heads, or 3 heads.

Visualizing this can help one appreciate the scope of possible outcomes and serves as a foundation for calculating probabilities. Knowing the sample space allows us to build probability functions that can forecast various event outcomes within this 'universe' of possibilities.

Binomial Distribution

The 'Binomial Distribution' is a specific probability distribution most often used when there are fixed, finite trials, each with two possible outcomes. It is perfect for scenarios like coin tosses or yes/no tests where each trial is independent of the others.

• The necessary parameters are the number of trials \( n \) and the probability of success \( p \) on each trial.
• In our coin flip example, we have 3 trials (flips), and since the head is twice as likely as tails, the probability of success, or getting a head, is \( \frac{2}{3} \).

Understanding Binomial Probability Formula

The probability of getting exactly \( k \) successes (heads) in \( n \) trials is described by the formula:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]In our case:

• \( \binom{3}{k} \) is the binomial coefficient which gives the number of ways to choose \( k \) trials out of \( n \).
• \( p^k \) accounts for the probability of getting \( k \) successes.
• \( (1-p)^{n-k} \) is for getting \( n-k \) failures.

This distribution helps us calculate the likelihood of different numbers of heads appearing in three tosses, such as the probability for 'at least two heads'.

Probability Density Function

A 'Probability Density Function' (PDF) provides a way of representing a probability distribution for a discrete or continuous random variable. In our context with coin tosses, the PDF is discrete because the number of heads you get in three tosses is finite and countable.

• The PDF describes probabilities for each possible outcome in the sample space produced by our experiment.
• These probabilities are calculated using the binomial distribution formula, ensuring they sum up to 1, adhering to the basic property of probability distributions.

Calculating Probabilities in Our Example

Here are the calculated probabilities using PDF for each outcome in our sample space after tossing the biased coin three times:

• \( P(0) = \frac{1}{27} \) for 0 heads.
• \( P(1) = \frac{2}{9} \) or \( \frac{6}{27} \) for 1 head.
• \( P(2) = \frac{4}{9} \) or \( \frac{12}{27} \) for 2 heads.
• \( P(3) = \frac{8}{27} \) for 3 heads.

A PDF provides a clear picture of how likely each result in the sample space is, guiding decisions and expectations for the outcomes of probabilistic experiments.