Problem 7 - Algebra and Trigonometry

Question

Five independent trials of a binomial experiment with probability of success \(p=0.7\) and probability of failure \(q=0.3\) are performed. Find the probability of each event. Exactly one success

Step-by-Step Solution

Answer

The probability of exactly one success is 0.02835.

Step 1: Identify the Parameters

In a binomial experiment, we have three main parameters: the number of trials \(n\), the probability of success in each trial \(p\), and the probability of failure \(q\). For this problem, we have \(n = 5\), \(p = 0.7\), and \(q = 0.3\).

Step 2: Write Down the Binomial Formula

The probability of having exactly \(k\) successes in \(n\) independent trials is given by the binomial formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Where \( \binom{n}{k}\) is the binomial coefficient.

Step 3: Calculate the Binomial Coefficient

For \(k = 1\), calculate the binomial coefficient \(\binom{5}{1}\):\[ \binom{5}{1} = \frac{5!}{1!(5-1)!} = 5 \]

Step 4: Apply the Binomial Formula

Substitute \(n = 5\), \(k = 1\), \(p = 0.7\), and \(q = 0.3\) into the binomial formula:\[ P(X = 1) = \binom{5}{1} (0.7)^1 (0.3)^{5-1} \]\[ P(X = 1) = 5 \times 0.7 \times (0.3)^4 \]

Step 5: Perform the Calculations

Evaluate the expression:\( (0.3)^4 = 0.0081 \)Then calculate:\[ P(X = 1) = 5 \times 0.7 \times 0.0081 = 0.02835 \]

Step 6: Write the Final Answer

The probability of having exactly one success in five trials is \(0.02835\).

Key Concepts

Binomial Coefficient

The binomial coefficient is a crucial part of the binomial probability formula. Think of it as a way to count how many different ways we can choose a subset of items from a larger set, where the order does not matter. In our context, it helps in determining how many ways we can arrange a certain number of successes among a given number of trials. For example, when we want to find the number of ways to have exactly one success in five trials, we use the binomial coefficient notation \(inom{n}{k}\), which is read as 'n choose k'. Here, \(n\) is the total number of trials (5 in our case), and \(k\) is the number of successes we're interested in (1 in this problem).To compute the binomial coefficient \(\binom{5}{1}\), we use the formula:

• \[ \binom{5}{1} = \frac{5!}{1!(5-1)!} = \frac{5 \times 4!}{1 \times 4!} = 5 \]

Factorials are simply the product of all positive integers up to that number, so \(5!\) is \(5 \times 4 \times 3 \times 2 \times 1\).This calculation tells us there are 5 different ways to have exactly one success in five trials.

Probability of Success

In a binomial experiment, the probability of success, denoted by \(p\), is the chance that a single trial results in a success. This value is fixed for every trial in the experiment. Knowing \(p\) is essential as it directly influences the outcome probabilities of different scenarios in a binomial setting. In our scenario, the probability of success \(p\) is given as 0.7. This means there is a 70% chance for each trial to result in a success. Alongside this, we should consider the probability of failure \(q\), which is calculated as \(q = 1 - p = 0.3\). This signifies there is a 30% chance of failure for each trial.When calculating the probability of exactly one success in five trials, we employ both \(p\) and \(q\). The binomial probability formula includes terms \(p^k\) for successes and \(q^{n-k}\) for failures:

• For one success: \((0.7)^1 = 0.7\)
• For four failures: \((0.3)^{4} = 0.0081\)

This balance between success and failure probabilities is what defines the shape of our binomial distribution.

Independent Trials

In a binomial experiment like ours, it's crucial to understand the concept of independent trials. Each trial is independent, meaning the outcome of one trial does not affect the outcomes of subsequent ones. This independence is what makes it possible to multiply probabilities across trials and use the binomial probability formula effectively.For instance, when discussing a sequence of trials where the outcome is either a success or failure, each trial should meet these conditions:

• The probability of success \(p\) remains constant (70% or 0.7 in our example).
• The trials are identical and conducted in the same way every time.

The independence ensures that the probability of achieving a certain number of successes (such as one success in our five trials) can be accurately computed without interference from previous outcomes.Thus, the assumption of independent trials is fundamental in applying the binomial model, and it assures that each trial is just like running the experiment anew under the same conditions.