Problem 66 - Algebra 1

Question

Find the probability of the event. Choosing the letter \(N\) from a bag that contains all 26 letters of the alphabet.

Step-by-Step Solution

Answer

The probability of selecting the letter 'N' randomly from a bag containing all the 26 letters of the English alphabet is \(\frac{1}{26}\)

Step 1: Define Total and Favorable Outcomes

Determine the number of total outcomes and favorable outcomes. The total number of outcomes is 26 because there are 26 different letters in the English alphabet. The desirable outcome, choosing the letter 'N', is just one. Therefore, we have one favorable outcome and 26 possible outcomes.

Step 2: Calculate the Probability

Using the formula for probability, which is favorable outcomes divided by total outcomes, we can evaluate the probability. Let's plug in our values into the formula. The probability \(P\), is given by \(P = \frac{favorable outcomes}{total outcomes}\). Plugging the values we get: \(P = \frac{1}{26}\)

Key Concepts

Favorable Outcome

In probability, a favorable outcome is the occurrence of the event you are trying to measure. In our example, the favorable outcome is drawing the letter 'N' from a bag containing all 26 letters of the alphabet. Simply put, when we talk about favorable outcomes, we're focusing on the specific result that we want.

Here's what you should remember about favorable outcomes:

• It is the specific result that you're interested in.
• In exercises dealing with probability, carefully identify what the event is.
• For the letter 'N' example, our favorable outcome count is 1 since 'N' is just one letter of the 26 available.

Grasping the idea of what makes an outcome favorable is essential for calculating probabilities correctly. Always ensure you're clear on what event you are considering to measure the probability for.

Total Outcomes

Total outcomes represent all the possible results that could occur in an experiment or scenario. In this example, the total outcomes are the total number of letters we could potentially draw from the bag, which is 26 (since the English alphabet consists of 26 letters).

Understanding total outcomes is crucial because:

• It is the denominator in the probability formula, representing all possible results.
• In a closed set, like the alphabet, ensuring you've accounted for all possibilities is critical to a valid probability measure.
• Knowing your total outcomes enables you to understand the scope of your experiment or scenario.

When calculating probabilities, always start by verifying the number of total outcomes to ensure accurate results.

Probability Formula

The probability formula is a fundamental tool for calculating the likelihood of an event. The formula is expressed as:

\[P = \frac{\text{favorable outcomes}}{\text{total outcomes}}\]This formula allows us to measure how likely an event is to occur.

To effectively use the probability formula, consider:

• Identifying your favorable outcomes—what you want to happen.
• Knowing your total outcomes—all possible results in the set.
• Plugging these values into the formula to solve for probability.

For example, if we're trying to find the probability of drawing 'N', we substitute 1 for favorable outcomes and 26 for total outcomes. Your solution is:
\[P = \frac{1}{26} \approx 0.0385\]This result means there is a 1 in 26 chance of picking 'N', or approximately a 3.85% chance. Use this basic formula to tackle a wide array of probability problems with confidence.