Solving Probability and Odds Problems in Basic Statistics: A Comprehensive Guide

Understanding and solving statistical problems, particularly those involving odds and probability, are essential for data analysis and decision-making processes. This guide will walk you through solving a specific problem that involves a bag containing red and blue discs. By the end, you will not only find the solution but also learn the steps and concepts involved in solving such problems.

The Problem

Let's consider the following problem: A bag contains 8 red discs and x blue discs. The odds against drawing a blue disc are 2:5. We need to find the value of x.

Step-by-Step Solution

To solve for the number of blue discs (x), let's break down the problem step by step:

Step 1: Understand the Odds Against Event

The odds against an event are given by the ratio of the number of unfavorable outcomes to the number of favorable outcomes. Here, the odds against drawing a blue disc are 2:5. This means that for every 2 unfavorable outcomes (drawing a red disc), there are 5 favorable outcomes (drawing a blue disc).

Step 2: Set Up the Total Number of Discs

The total number of discs in the bag is:

(8 text{ red discs} x text{ blue discs} 8 x)

The number of unfavorable outcomes (drawing a red disc) is 8, and the number of favorable outcomes (drawing a blue disc) is x.

Step 3: Express the Odds Formally

From the given odds, we can set up the following mathematical equation to represent the ratio of unfavorable to favorable outcomes:

(frac{text{Number of unfavorable outcomes}}{text{Number of favorable outcomes}} frac{2}{5})

Substituting the number of red discs and blue discs, we get:

(frac{8}{x} frac{2}{5})

Step 4: Cross-Multiply to Solve for x

To solve for x, we cross-multiply:

8 times; 5 2 times; x

This simplifies to:

40 2x

Step 5: Solve for x

Dividing both sides by 2:

(x frac{40}{2} 20)

Conclusion: The value of x is boxed{20}.

Additional Insights

Let's delve deeper into understanding the solution and explore the concepts used:

Probability and Odds in Detail

Probability (Pr): The probability of drawing a red disc (Pr) is given by the ratio of the number of red discs to the total number of discs:

(Pr frac{8}{8 x})

Probability of Blue Disc (Pb): The probability of drawing a blue disc (Pb) is:

(Pb frac{x}{8 x})

The odds against drawing a blue disc are given by:

(frac{[1 - Pb]}{Pb} frac{2}{5})

Solving this, we get:

(frac{1 - frac{x}{8 x}}{frac{x}{8 x}} frac{2}{5})

(frac{8 x - x}{x} frac{2}{5})