Problem 15 - Algebra 1

Question

Find the odds of randomly choosing the indicated letter from a bag that contains the letters in the name of the given state. A; NEBRASKA

Step-by-Step Solution

Answer

The odds of randomly choosing the letter 'A' from a bag that contains the letters in the name 'NEBRASKA' is \( \frac{3}{8} \).

Step 1: Count the total number of letters

The given word is 'NEBRASKA'. Count the total number of letters in the word 'NEBRASKA'. The total count is 8.

Step 2: Count the number of times letter 'A' appears

Now, count the number of times the letter 'A' appears in 'NEBRASKA'. The count is 3.

Step 3: Calculate the probability

Now having the total number of letters in the word 'NEBRASKA' which is 8 and the number of times the letter 'A' appears which is 3, the odds of randomly selecting this letter can be calculated as the number of 'A' letters over the number of total letters. This gives \( \frac{3}{8} \).

Key Concepts

Calculating Odds

Odds are a fundamental concept in probability, used to measure the likelihood of a particular outcome. They are often confused with probability, but there's a distinction between the two. While probability measures the chance of a certain event happening, odds compare the likelihood of an event occurring to the likelihood of it not occurring.

To calculate the odds in favor of an event, you divide the number of favorable outcomes by the number of unfavorable outcomes. In our exercise, choosing the letter 'A' from the name 'NEBRASKA,' we first determined there are 3 'A's (favorable outcomes) and 5 other letters (unfavorable outcomes). Hence the odds in favor of picking an 'A' would be expressed as 3 to 5 (or simply '3:5'). It is important to note that odds can be represented as ratios or as decimals.

Probability Problems

Probability problems require us to calculate the likelihood of an event occurring, which can often incorporate algebraic concepts and fractions. These problems typically involve certain steps, including defining the sample space, determining favorable outcomes, and using formulas to find the probability.

In the case of our exercise, the sample space consists of the letters in the word 'NEBRASKA.' After identifying the event of interest (selecting the letter 'A'), we calculated its probability by dividing the number of times 'A' appears in the name (favorable outcomes) by the total number of letters (sample space size). The probability of randomly choosing an 'A' from 'NEBRASKA' was found to be \( \frac{3}{8} \). Always remember that the probability value will range between 0 and 1, where 0 indicates an impossibility, and 1 indicates a certainty.

Algebraic Fractions

Algebraic fractions, much like numerical fractions, involve numerators and denominators but they can include variables, exponents, and algebraic expressions. In solving probability problems, we often deal with numerical fractions, which is the ratio of two integers.

In our exercise with 'NEBRASKA,' the probability of randomly selecting the letter 'A' resulted in the fraction \( \frac{3}{8} \). Understanding fractions is crucial in probability since they describe the proportion of the favorable outcomes to all possible outcomes. This fraction simplifies the representation of the data and provides clarity on the likelihood of an event. When dealing with more complex algebraic fractions in probability, you might need to simplify expressions or solve equations to find the probabilities you're interested in.