Problem 61 - Algebra and Trigonometry

Question

On The Price Is Right, there is a game in which a bag is filled with 3 strike chips and 5 numbers. Let's say that the numbers in the bag are \(0,1,3,6,\) and \(9 .\) What is the probability of selecting a strike chip or the number \(1 ?\)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{2} \).

Step 1: Identify Total Number of Items in the Bag

First, calculate the total number of items in the bag. There are 3 strike chips and 5 numbers, so the total number of items is 3 + 5 = 8.

Step 2: Determine the Number of Favorable Outcomes

Identify how many items match the criteria of selecting a strike chip or the number 1. There are 3 strike chips and 1 number 1, so there are 3 + 1 = 4 favorable outcomes.

Step 3: Calculate the Probability

The probability \( P \) of an event is given by the formula \( P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \). Here, the probability is \( P = \frac{4}{8} = \frac{1}{2} \).

Key Concepts

Strike Chips

In the game described from The Price Is Right, strike chips play a crucial role. These strike chips are among the items in the bag, and each chip represents an unfavorable outcome when players draw them. In this scenario, there are 3 strike chips in the bag, which means if you pull out a strike chip, it's typically something you don't want to happen. This introduces an element of risk into the game, making it more exciting and unpredictable.
Strike chips essentially stand in the way of a positive outcome, so understanding their role is vital when calculating the probability of drawing something else, like a number.
Knowing the exact count of strike chips helps in accurately determining the chances of any particular event.

Favorable Outcomes

Favorable outcomes are the specific results you are interested in when calculating probabilities. In our exercise, favorable outcomes include drawing either a strike chip or the number 1.
To identify favorable outcomes:

• List out all the outcomes you consider favorable (here, it's drawing a strike chip or the number 1).
• Count how many such outcomes exist in the set of all possible outcomes (which in this case is 3 strike chips + 1 number 1 = 4 favorable outcomes).

Recognizing and counting favorable outcomes is essential for calculating the probability of any event accurately.

Probability Formula

Probability is a measure of how likely an event is to occur. It's calculated using the probability formula:
\[ P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \]
Here's the step-by-step application of the probability formula in our exercise:

• First, identify the number of favorable outcomes (in this case, the count is 4).
• Next, find the total number of possible outcomes (the total items in the bag, which is 8).
• Finally, divide the number of favorable outcomes by the total number of possible outcomes.

Using these steps, the probability calculation would be \[ P = \frac{4}{8} = \frac{1}{2} \] indicating a 50% chance of drawing either a strike chip or the number 1.

Total Possible Outcomes

The concept of total possible outcomes is a fundamental part of calculating probability. Total possible outcomes refer to all possible results that can occur in a given scenario.
In our exercise, we need to consider everything that can be drawn from the bag. To find the total number of possible outcomes:

• Add up every individual item that could be drawn (3 strike chips + 5 numbers).

For this example, the calculation is 3 + 5 = 8, meaning there are 8 possible outcomes when you draw one item from the bag.
Understanding the total possible outcomes helps give context to your probability calculation and ensures that all potential scenarios are considered in your analysis.