Question

If a letter is selected at random from the letters of the word LOGARITHMS, then what is the probability that it will be a consonant? (1) \(3 / 10\) (2) \(7 / 10\) (3) \(1 / 10\) (4) \(4 / 10\)

Short answer

Answer: (2) \(7 / 10\)

Step-by-step solution

List all consonants in the word LOGARITHMS

First, we will list all the consonants present in the word "LOGARITHMS". The consonants are L, G, R, T, H, M, and S.

Count the total number of consonants

Now, let's count the total number of consonants. There are 7 consonants in the word "LOGARITHMS": L, G, R, T, H, M, and S.

Count the total number of letters

Next, we will count the total number of letters in the word "LOGARITHMS". There are 10 letters in total: L, O, G, A, R, I, T, H, M, and S.

Calculate the probability

Finally, we will calculate the probability of selecting a consonant at random from the word "LOGARITHMS". The probability is the ratio of the number of consonants to the total number of letters, which is \(\frac{7}{10}\). So, the correct answer is (2) \(7 / 10\).

Key concepts

Consonant Probability

Understanding the probability of drawing a consonant from a set of letters is an intriguing application of mathematics. Let's take a closer look with the example of the word 'LOGARITHMS'. To calculate consonant probability, firstly, you need to distinguish consonants from vowels within the context of the given word.

As highlighted in the step-by-step solution, the consonants in 'LOGARITHMS' are L, G, R, T, H, M, and S. By establishing that there are 7 consonants, you next need to consider the total number of letters in the word, which is 10. Probability, at its core, is about ratios. The consonant probability is simply the ratio of the number of consonants to the total number of letters, expressed as \(\frac{7}{10}\). This means that if you were to select a letter at random, there is a 70% chance it would be a consonant—a significant majority.

Understanding this concept is vital, not just for this problem but for recognizing patterns in language and applying probability to linguistics. Statistical literacy often starts with such real-world examples, making abstract concepts like probability more tangible.

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, combination, and permutation of sets. It serves as the building blocks for calculating probabilities. When solving problems like the one presented, combinatorics helps in enumerating possible outcomes and organizing them efficiently. In the given exercise, we implicitly used combinatorial reasoning to determine the number of ways we can choose consonants from the set of all letters in 'LOGARITHMS'.

Counting Consonants

Identifying the 7 consonants among the 10 letters requires you to utilize combinatorial thinking. What matters here is the number of distinct elements – in this case, different consonants – and not the order in which they appear.

Combinatorics isn't used explicitly in this problem, but understanding its principles can greatly help in more complex probability challenges. For instance, if we were to calculate the probability of selecting 2 consonants in a row without replacement, combinatorial formulas would be required to determine all possible outcomes.

Mathematical Probability

Mathematical probability is the measure of the likelihood of an event to occur, represented as a number between 0 and 1, where 0 indicates impossibility and 1 denotes certainty. The concept is extensively used in a range of disciplines, from gambling to weather forecasting.

In the context of our exercise, the event is selecting a consonant from the letters in 'LOGARITHMS'. Probability is calculated by dividing the number of favorable outcomes (in our case, consonants) by the total number of possible outcomes (all letters in the word). Thus, the mathematical probability that the event of selecting a consonant will occur is expressed as \(\frac{7}{10}\).

Relevance of Probability

The relevance of mathematical probability extends beyond academics; it helps us make informed decisions in uncertain situations. By understanding probability, one can better assess risks and chances in daily life, from simple decisions like taking an umbrella on a cloudy day, to more complex ones like investing in the stock market. Mathematical probability is a fundamental tool that aids in structuring the way we think about uncertainty.