Question

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

Short answer

The odds of picking 'N' from a bag containing the letters of the word 'PENNSYLVANIA' is 1/6.

Step-by-step solution

Identify Letter and Total Count

First, identify the number of occurrences of 'N' and the total number of letters in 'PENNSYLVANIA'. Upon counting, we find there are 2 'N's and 12 letters in total in 'PENNSYLVANIA'.

Calculate Odds

The probability is calculated by dividing the number of ways to successfully choose the letter of interest by the total number of outcomes. Here, successful ways equals to number of N's which is 2 and total outcomes equals to total number of letters which is 12. So, the odds are two in twelve or, equivalently, one in six.

Simplify the Result

The probability can be simplified to lowest terms for better understanding. Here, divide both the numerator and the denominator of the fraction by their greatest common divisor, which is 2. Hence, the odds reduce to 1/6. This means there's a one in six chance of randomly picking 'N' from the letters of the word 'PENNSYLVANIA'.

Key concepts

Odds Calculation

When it comes to understanding probability, one of the starting points is learning how to calculate the odds of an event. Odds represent the likelihood that a specified event will occur. To put it simply, they express the ratio of the number of ways an event can happen to the number of ways it cannot.

In our example with the state name 'PENNSYLVANIA', we are trying to find the odds of randomly selecting the letter 'N'. To do this, we first count how many times 'N' appears, which is twice, and then we count the total number of letters in the word, which is twelve.

The mathematical expression for odds is usually given in the form of 'successes to failures'. In this case, we consider a 'success' to be choosing an 'N', and a 'failure' to be choosing any other letter. Thus, the initial odds are 2 successes to 10 failures (since there are 12 letters in total and 2 of them are 'N'). Simplifying this ratio, we express the odds as 1 to 5, which is the simplified form of odds calculation.

Probability Simplification

Probability simplification is an essential step to make the results more understandable. By breaking down probabilities into their simplest form, it becomes easier to grasp the likelihood of an event occurring. This concept is closely tied with the math practice of simplifying fractions.

To simplify the probability, you identify the greatest common divisor (GCD) of both the numerator (the number of successful outcomes) and the denominator (the total number of possible outcomes) and then divide both by that number. In the context of our 'PENNSYLVANIA' example, the initial odds were two in twelve, or \(\frac{2}{12}\). The greatest common divisor of 2 and 12 is 2, which means we can divide both numbers by 2, thus simplifying the odds to \(\frac{1}{6}\).

The result is not only easier to read, but it also helps when comparing probabilities. A simplified fraction paints a clear picture; in this case, there is a one in six chance of drawing an 'N' from a bag containing all the letters of 'PENNSYLVANIA'.

Probability of an Event

The probability of an event is a measure of the chances that the event will occur. It's a basic concept in mathematics and is especially important in statistics, gaming, and various applications in everyday life.

Mathematically, the probability is expressed by a number between 0 and 1, where 0 indicates that the event cannot happen, and 1 indicates certainty that the event will happen. To find the probability, one must consider all possible outcomes and how many of them count as a 'success'.

In our exercise example, randomly choosing the letter 'N' from 'PENNSYLVANIA' has a probability which, after simplification, is \(\frac{1}{6}\). This fraction means that if you were to pick a letter out of the bag 6 times (with replacement, so the total count of letters stays the same each time), on average, you'd select the letter 'N' once. This fraction points to the fractional likelihood of the desired outcome occurring within a series of attempts.