Problem 62 - Calculus for Biology and Medicine

Question

A random experiment consists of flipping a biased coin with probability \(0.3\) of heads until the first time heads appears. Find the probability that heads appears for the first time on the fifth trial.

Step-by-Step Solution

Answer

The probability is approximately 0.072.

Step 1: Understanding the Problem

We are given a biased coin with a probability of heads being 0.3. The problem involves finding the probability of getting the first head on the fifth flip.

Step 2: Formula for Geometric Distribution

This scenario is an example of a geometric distribution, which models the number of trials until the first success. The probability mass function (PMF) for the first success on the nth trial is given by \( P(X=n) = (1-p)^{n-1} \, p \), where \( p \) is the probability of success (head) in one trial.

Step 3: Apply the Formula

Here, \( p = 0.3 \) and the first head (success) should occur on the 5th trial. Substitute into the formula: \( P(X=5) = (1-0.3)^{5-1} \times 0.3 \).

Step 4: Calculate the Probability

Simplify the expression: \( (1-0.3) = 0.7 \). Therefore, \( P(X=5) = 0.7^4 \times 0.3 \). Calculate \( 0.7^4 \) and then multiply by 0.3 to get the probability.

Step 5: Final Computation

Compute \( 0.7^4 \approx 0.2401 \) and then \( 0.2401 \times 0.3 = 0.07203 \). Thus the probability that the first head appears on the fifth trial is approximately \( 0.07203 \).

Key Concepts

Probability

In probability, we often deal with the chance of an event occurring. Probability is a measure that quantifies the likelihood of events.
It ranges from 0 to 1, where 0 indicates impossibility and 1 indicates certainty.

• A probability of 0.5 indicates a fifty-fifty chance.
• When you flip a fair coin, the probability of getting heads is 0.5.

In the context of this exercise, we consider a biased coin, where heads appear with a probability of 0.3.
This predetermined probability is crucial in calculating outcomes and understanding how likely it is for a particular event, like a head appearing on a coin flip, to occur.
Understanding the basics of probability helps us make sense of how often we can expect certain results from random processes.

Biased Coin

A biased coin is simply one that does not have equal probabilities for heads and tails. In the exercise at hand, the coin has a 30% chance of landing on heads, meaning it's biased.

• A fair coin would have a 50% chance for heads and a 50% chance for tails.
• The biased nature of the coin influences how we calculate probabilities for sequences of tosses, as not every coin flip is created equal in terms of outcome potential.

This bias towards a particular result (in this case, tails) must be factored into our calculations.
It affects the expectation of how long it takes for certain events to happen, like flipping until a head shows up for the first time.
Understanding bias in probability exercises is critical for correctly applying mathematical models like the geometric distribution.

Probability Mass Function

A probability mass function (PMF) is used to describe the probability distribution of a discrete random variable.
In simpler terms, it gives probabilities of different possible outcomes for a discrete random event.
In our problem, the PMF is expressed using the formula: \[P(X=n) = (1-p)^{n-1} \times p\] where \( p \) is the probability of success on a single trial and \( n \) represents the trial on which the first success occurs.

• This formula specifically applies to the geometric distribution, which models the number of trials up to and including the first success.
• In the context of our exercise, this function helps us find out how likely it is for the first head to appear on any given trial, such as the fifth trial.

Understanding how to interpret and use a PMF is essential for solving problems related to discrete random events.

Trials

Trials refer to each independent repetition of a random experiment, like flipping the coin repeatedly until a head shows up.
Each trial is treated as an individual event that can either succeed (yield a head) or fail (yield a tail).

• In problems like this, trials help us analyze how often we need to repeat an experiment to achieve our desired outcome.
• In this case, our goal is to find out how likely it is for the first head to appear specifically on the fifth trial.

By understanding what each trial represents, we can correctly apply probabilistic formulas and theories to predict the outcomes of these repeated attempts.
This concept is significant in probabilistic models where the outcome isn't guaranteed and relies on repetition and chance.