Problem 11 - 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. At least two successes

Step-by-Step Solution

Answer

The probability of at least two successes is approximately 0.96922.

Step 1: Understanding the Binomial Distribution

A binomial distribution is characterized by a fixed number of trials, each trial having two possible outcomes (success or failure), and each trial being independent. The probability of success in each trial is constant. Here, we have 5 trials with a probability of success, \( p = 0.7 \), and a probability of failure, \( q = 0.3 \).

Step 2: Define the Event

We need to calculate the probability of getting at least 2 successes in 5 trials. This is the sum of probabilities of having 2, 3, 4, or 5 successes.

Step 3: Calculate Individual Probabilities using Binomial Formula

The probability of getting exactly \( k \) successes in \( n \) trials is given by the binomial formula:\[P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\]Calculate for \( k=2 \), \( k=3 \), \( k=4 \), and \( k=5 \) with \( n=5 \).

Step 4: Calculate Probability for 2 Successes

\[P(X=2) = \binom{5}{2} (0.7)^2 (0.3)^3 = 10 imes 0.49 imes 0.027 = 0.1323\]

Step 5: Calculate Probability for 3 Successes

\[P(X=3) = \binom{5}{3} (0.7)^3 (0.3)^2 = 10 imes 0.343 imes 0.09 = 0.3087\]

Step 6: Calculate Probability for 4 Successes

\[P(X=4) = \binom{5}{4} (0.7)^4 (0.3)^1 = 5 imes 0.2401 imes 0.3 = 0.36015\]

Step 7: Calculate Probability for 5 Successes

\[P(X=5) = \binom{5}{5} (0.7)^5 (0.3)^0 = 1 imes 0.16807 imes 1 = 0.16807\]

Step 8: Summation of Desired Probabilities

Add the probabilities for 2, 3, 4, and 5 successes:\[P(X \geq 2) = 0.1323 + 0.3087 + 0.36015 + 0.16807 = 0.96922\]

Step 9: Conclusion

The probability of having at least 2 successes in 5 trials is approximately 0.96922.

Key Concepts

Probability of Success

The probability of success is a fundamental concept in probability theory, particularly within the framework of a binomial distribution. It refers to the likelihood that a single trial in a series will result in a desired outcome. In a binomial experiment, which involves a fixed number of trials with two possible results (success or failure), this probability remains constant across all trials.
In our example, you consider a binomial experiment conducted over five independent trials. Here, the probability of success on each trial is given as 0.7. This means that each individual trial has a 70% chance of hitting the intended target, so to speak.

• This probability is usually denoted by the letter \( p \).
• The probability of failure (not succeeding) is denoted by \( q \).
• In our case, \( q = 1 - p = 0.3 \).

Remember that both \( p \) and \( q \) add up to 1, ensuring that one of these outcomes will always occur.

Independent Trials

In a binomial distribution, independent trials are key. Each trial must not affect the outcome of another, meaning the result of one trial cannot change the probabilities of the next. This independence ensures that we can treat each trial as a fresh start, maintaining a constant probability of success throughout.
In the exercise, you are dealing with five trials - each counted as independent. This is crucial as it ensures the calculations for probabilities from each trial remain accurate and untainted by previous results.

• Independence allows us to use the binomial formula, which requires the probabilities to stay fixed across trials.
• Because trials are independent, past successes or failures don't influence future ones.

Maintaining independence is essential to using the binomial model effectively, ensuring the validity of your calculations.

Binomial Formula

The binomial formula is a handy tool that allows you to calculate the probability of achieving a specific number of successes in a given number of trials. It is expressed mathematically for a binomial distribution as follows: \[P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\]
Here's how each component of the formula fits in:

• \( \binom{n}{k} \) represents the number of ways to choose \( k \) successes in \( n \) trials.
• \( p^k \) denotes the probability of success raised to the power of \( k \), accounting for \( k \) successes.
• \((1-p)^{n-k}\) is the probability of failure raised to the power of \( n-k \), covering the remaining trials leading to failure.

In the exercise, this formula helps you compute the probability of achieving exact numbers of successes (like 2, 3, 4, or 5) within the five trials. By summing these probabilities, you find the likelihood of having at least two successes, arriving at approximately 0.96922.