Problem 22 - Algebra 2

Question

A calculator contains four batteries. With normal use, each battery has a 90\(\%\) chance of lasting for one year. Find the probability that all four batteries will last a year.

Step-by-Step Solution

Answer

The probability that all four batteries will last a year is approximately 65.61%.

Step 1: Identify the Individual Probability

First, we need to identify the individual probability of each battery lasting one year. This has been given as 90%, or in decimal form as 0.90.

Step 2: Multiply Individual Probabilities

Since these are independent events, we multiply the individual probabilities. For four batteries, we'll multiply the individual probability four times.

Step 3: Calculate the Probability

The final step is to calculate the result of the multiplication. \(0.90 * 0.90 * 0.90 * 0.90 = 0.6561 or 65.61%\)

Key Concepts

Independent Events

Understanding independent events is crucial when dealing with probabilities like in our battery example. Two or more events are independent when the outcome or occurrence of one does not affect the outcome or occurrence of another.
This means the chance of each event happening is the same, no matter what happens to other events. In the context of our calculator’s batteries, each battery has a 90% chance of lasting a year.
The lifespan of one battery does not impact the lifespan of another. Recognizing this independence allows us to combine probabilities correctly using simple multiplication.

Probability Calculation

Calculating probabilities involves taking individual event probabilities and combining them to find the probability of a sequence of events. When you know the probability of a single event — for instance, one battery lasting a year is 90% — you can calculate multiple events’ probabilities by multiplying.
It's like piecing together a puzzle, where each piece's likelihood (or probability) contributes to the overall picture. For battery success, we relate the power of each contributing factor to get the joint probability.

Decimal Conversion

Decimal conversion is a simple yet essential step in probability calculations. It converts percentage probabilities into a decimal form that is easier to handle in calculations.
For example, the 90% chance for a battery lasting turns into 0.90. By converting percentages, calculations become straightforward, particularly when multiplying probabilities, as you don't have to manage cumbersome percentages.
It’s as simple as moving a decimal point two places to the left — a tiny action with big impact in math operations.

Multiplication Rule in Probability

The multiplication rule is a fundamental principle for calculating the probability of independent events. This rule only works when events do not influence each other.
By multiplying their individual probabilities, you determine the likelihood of all outcomes occurring together. With our battery problem, we take the 0.90 chance for each battery and multiply it four times, getting a combined probability of around 65.6%.
This rule enables us to handle complex situations with ease, providing a simple method to determine probabilities for events happening in sequence.