Question

If a letter is selected at random from the letters of the word 'FOUNDATION', what is the probability of the letter selected being a repeated letter? (1) \(\frac{1}{10}\) (2) \(\frac{1}{5}\) (3) \(\frac{2}{5}\) (4) \(\frac{1}{2}\)

Short answer

Answer: The probability of selecting a repeated letter from the word 'FOUNDATION' is \(\frac{2}{5}\).

Step-by-step solution

Identify the given word and its distinct letters

In this case, the given word is 'FOUNDATION.'

Count the total number of letters in the word and identify the repeated letters

The total number of letters in the word 'FOUNDATION' is 10. The repeated letters are 'O' and 'N.'

Count the number of repeated letters

In the word 'FOUNDATION,' - 'O' appears 2 times - 'N' appears 2 times Total number of repeated letter occurrences = 2 + 2 = 4

Use the probability formula to find the probability of selecting a repeated letter

The probability formula is: Probability = \(\frac{\text{Number of repeated letter occurrences}}{\text{Total number of letters in the word}}\) Plugging in the numbers, we get: Probability = \(\frac{4}{10}\)

Simplify the fraction

Simplify the fraction \(\frac{4}{10}\) by dividing the numerator and the denominator by their greatest common divisor (which is 2): Probability = \(\frac{2}{5}\) The probability of selecting a repeated letter from the word 'FOUNDATION' is \(\frac{2}{5}\). Hence, the correct answer is (3) \(\frac{2}{5}\).

Key concepts

Repeated Letters

In the context of probability exercises involving words, "repeated letters" are those letters that appear more than once in a word. To solve the given problem, you start by identifying all unique letters in the word and their frequency. In the word "FOUNDATION," the letters 'O' and 'N' are repeated, each appearing twice. Understanding which letters repeat is crucial when calculating the probability of selecting one of these letters from the entire word. Analyzing the repetition helps set up the parameters for your probability calculation.

Fraction Simplification

Fraction simplification is a fundamental step in probability problems, especially after computing an initial probability fraction. When you get a fraction like \(\frac{4}{10}\), simplifying it makes understanding and further calculations easier. You simplify by finding the greatest common divisor (GCD) of the numerator and denominator. Here, the GCD for 4 and 10 is 2. By dividing both terms by 2, the fraction becomes \(\frac{2}{5}\). Simplifying fractions ensures you work with the simplest and most easily interpretable form, which is necessary for clear communication of your answer.

Probability Formula

The probability formula is a key concept in this exercise. It defines how you calculate the likelihood of an event happening. The formula used here is: \[\text{Probability} = \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}}.\]In the "FOUNDATION" exercise, the successful outcomes are represented by the occurrences of repeated letters, and the total outcomes are the total number of letters. Plugging the numbers into the formula provides \(\frac{4}{10}\), which after simplification, shows the probability of randomly selecting a repeated letter from "FOUNDATION." Being familiar with and applying the probability formula allows you to solve various types of probability problems effectively.

Counting Principles

Counting principles are essential when determining probabilities in word-based problems. These principles help determine the total number of outcomes and the successful (or target) outcomes. In the given exercise, each letter in the word "FOUNDATION" represents an outcome. The word contains ten letters, giving ten possible outcomes. Counting principles also apply when assessing repeated letters. The process involves simply counting how often each letter appears. Understanding these principles ensures accuracy when calculating the likelihood of events, setting a strong foundation for applying the probability formula accurately.