Question

What is the probability that Daryl will pass two out of the three tests? Round your answer to the nearest whole percentage. (A) 30% (B) 46% (C) 52% (D) 190%

Short answer

The probability that Daryl will pass two out of the three tests is approximately \(38%\), which is closest to answer choice (B) \(46%\).

Step-by-step solution

Use the binomial formula to calculate the probability

We will use the binomial probability formula to find the probability of passing 2 out of 3 tests, assuming that Daryl has a 50% chance of passing each test: \[P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\] where \(n\) is the total number of tests, \(k\) is the number of successful tests, \(p\) is the probability of success on a single test, and \(\binom{n}{k}\) is the binomial coefficient that gives us the number of possible combinations of \(k\) successful tests out of \(n\). In our problem, we have \(n = 3\), \(k = 2\), and \(p = 0.5\).

Calculate the binomial coefficient

We will calculate the binomial coefficient \(\binom{3}{2}\), which gives us the number of ways of choosing 2 successful tests out of 3: \[\binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3!}{2!1!} = \frac{3\times2\times1}{(2\times1)(1)} = 3\]

Calculate the probability

Now we will use the binomial probability formula to calculate the probability of passing 2 out of 3 tests: \[P(X=2) = \binom{3}{2} 0.5^2(1-0.5)^{3-2} = 3\times0.5^2\times0.5^1 = 3\times0.25\times0.5 = 0.375\]

Round the probability to the nearest whole percentage

To round the probability to the nearest whole percentage, we will multiply by 100 and round the result: \[0.375\times100 = 37.5\] Rounding to the nearest whole percentage, we get 38%. Therefore, the probability that Daryl will pass two out of the three tests is approximately 38%, which is closest to answer choice (B) 46%.

Key concepts

Understanding the Binomial Formula

The binomial formula is a valuable tool in probability, especially when dealing with situations where there are two possible outcomes, like passing or failing a test. To apply this formula, we calculate the probability of a certain number of "successes" (which in Daryl’s case, is passing tests) within a fixed number of trials (the total number of tests). The formula is as follows:\[P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\]Here:

• \(n\) represents the total number of trials or tests
• \(k\) indicates the number of successful trials you're interested in
• \(p\) is the probability of success on a single trial
• \(1-p\) is the probability of failure on a single trial

This formula allows us to calculate how probable it is to achieve exactly a certain number of successes.

Explaining the Binomial Coefficient

The binomial coefficient is a central part of the binomial formula and can be written as \(\binom{n}{k}\). It represents the number of ways to choose \(k\) successes in \(n\) trials.The formula to calculate the binomial coefficient is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]This notation \(!\) is called a "factorial," which means multiplying all positive integers up to that number. For example, \(3!\) is \(3 \times 2 \times 1 = 6\).In Daryl's exercise, we calculated the binomial coefficient \(\binom{3}{2}\). This step is crucial as it tells us there are 3 ways to have 2 passes out of 3 tests.

Rounding Percentages Accurately

Once you have your probability result, often expressed as a decimal, you may need to convert it to a percentage. This requires multiplying by 100. An important final step is to round the percentage to the nearest whole number.For example, if the probability is 0.375, we multiply by 100 to convert it to a percentage:\[0.375 \times 100 = 37.5\]Rounding 37.5 to the nearest whole number gives us 38%. Rounding is essential because it presents a cleaner and more approximate value that’s easier for quick interpretation.

Simplifying Mathematical Calculations

Performing calculations, especially with probabilities, can seem daunting at first. Breaking down each mathematical step into smaller parts makes it more manageable.For instance, while solving Daryl's problem:

• First, find the binomial coefficient \(\binom{3}{2}\) by calculating \(\frac{3!}{2!1!}\).
• Next, compute \(0.5^2 = 0.25\) and \(0.5^1 = 0.5\).
• Then, multiply these together with the binomial coefficient: \(3 \times 0.25 \times 0.5 = 0.375\).

Once the basic calculations are clear, putting them together to solve the problem becomes straightforward. Practice simplifying each component of the formula separately, and the overall solution will follow logically.