Problem 13 - 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. S; MISSISSIPPI

Step-by-Step Solution

Answer

The odds of picking a 'S' out of the word 'MISSISSIPPI' are 4:7.

Step 1: Finding the Total Number of Letters

Count each individual letter in the name 'MISSISSIPPI'. 'MISSISSIPPI' contains 11 letters in total.

Step 2: Finding the Number of Occurrences of 'S'

Count how many times 'S' appears in 'MISSISSIPPI'. 'S' appears 4 times.

Step 3: Calculating the Odds

Calculate the odds by dividing the number of occurrences of 'S' by the total number of letters. This can be represented as a ratio (Number of 'S' : Remaining letters ). Hence, the odds of drawing an 'S' are 4 out of 11 or 4 : 7.

Key Concepts

Understanding Probability

Probability is fundamental in understanding the likelihood of an event occurring. When you're fishing for one particular letter out of a bag full of various letters, you're dealing with a classic scenario in probability. Specifically, you calculate the probability measure to evaluate the chance of drawing the letter 'S' from a bag with the word 'MISSISSIPPI'.

To do this, you start by identifying the total number of outcomes, which is simply the total count of letters in the bag. In our case, 'MISSISSIPPI' consists of 11 letters. The next step is to determine the number of times the event you are interested in, namely drawing an 'S', could happen. This word has 4 'S's. The probability, therefore, is the fraction \( \frac{4}{11} \) which represents the ratio of the favorable outcomes to the total possible outcomes.

Understanding this concept of probability is crucial, not just for homework exercises, but for making informed decisions in real life, where analyzing risks and chances often plays a vital role.

Ratio and Proportion

While it might seem similar to probability, the concept of ratio and proportion provides a different perspective. It's all about comparing quantities. When we say the odds of drawing an 'S', we're actually discussing a ratio: the number of 'S's to the number of non-'S' letters.

In our case, the ratio of 'S's (4) to non-'S' letters (7) is written as 4:7. It is key to understand that this ratio does not reflect the probability but rather the relative size of one group to another. Proportions further extend this idea, indicating that two ratios are equivalent. If we had a larger bag with twice as many of each letter in 'MISSISSIPPI', the ratio of 'S's to non-'S's would still be 4:7, showing a proportional relationship.

The Role of Combinatorics

Combinatorics, the field of mathematics concerned with counting, comes into play when analyzing probability and ratios in more complex scenarios. It helps us figure out how many ways we can arrange or combine items without actually having to list all possible ways. When dealing with these letters, for instance, combinatorics could tell us how many different 5-letter combinations could be created from 'MISSISSIPPI'.

For simpler situations like our exercise, combinatorics confirms there is only one way to draw an 'S' each time but affirms that there are many ways to combine the letters into unique strings. This combinatorial outlook would be significantly more important when calculating probabilities of compound events, where the arrangement or selection order of these letters plays a part in the overall outcome.