Problem 51 - Algebra and Trigonometry Real Mathematics, Real People

Question

A random number generator selects three numbers from 1 through 10. Find the probability of the event. All three numbers are even.

Step-by-Step Solution

Answer

The probability of all three numbers being even is 0.125

Step 1: Identify even numbers from 1 to 10

Firstly, identify the even numbers in the range from 1 to 10. These are 2, 4, 6, 8, and 10. There are 5 even numbers in total.

Step 2: Calculate probability for each draw

As the numbers are being drawn with replacement (implied by the scenario), the probability of drawing an even number remains constant throughout all three draws. Compute this probability, which is \(\frac{Number Of Even Numbers}{Total Numbers}\) i.e., \(\frac{5}{10}\) or 0.5.

Step 3: Calculate joint probability of three draws

Next, calculate the joint probability of drawing an even number in all three draws. This is the product of the probabilities of each individual draw: \(Probability = (0.5) * (0.5) * (0.5) = 0.125\).

Key Concepts

Even Numbers

Even numbers are integers that can be divided evenly by 2. These are numbers like 2, 4, 6, 8, 10, and so on. In the context of the exercise, we are looking at even numbers between 1 and 10, which include 2, 4, 6, 8, and 10.

• Even numbers end with 0, 2, 4, 6, or 8.
• They are part of the sequence that includes every second number starting from 2.
• For our specific exercise, we identified all even numbers in the given range, which were necessary to calculate probabilities.

Understanding even numbers is crucial because it helps in calculating the probability of an event where we are interested only in even outcomes.

Random Number Generator

A random number generator (RNG) is a tool that generates a sequence of numbers that has no discernible pattern. In our exercise, the RNG selects numbers between 1 and 10.

• Randomness ensures that every number has an equal chance to be picked, regardless of past events.
• It is crucial in probability exercises to simulate the randomness needed for fair probability calculation.
• RNGs are used in various fields, from computer simulations to statistical sampling.

In our situation, we used an RNG to pick numbers independently and with replacement, meaning the process is entirely random each time.

Joint Probability

Joint probability refers to the likelihood of two or more events happening at the same time. In our example, it's the probability of selecting an even number in three consecutive draws.

• When calculating joint probability, you multiply the probability of individual events if they are independent.
• If each event has a probability of 0.5 (as calculated), the joint probability for three events is oinspruct{\(0.5^3 = 0.125\)}.
• This approach helps measure the combined likelihood of all events occurring together.

It's important in scenarios where multiple conditions need to be met simultaneously, such as our goal of all numbers being even.

Events with Replacement

The term 'with replacement' means that each event or trial does not affect the others, maintaining equal probabilities throughout.

• For each selection by the random number generator, the number is placed back, or 'replaced', keeping the number pool constant.
• This ensures the probability remains unchanged, calculated as oinspruct{\(\frac{5}{10} = 0.5\)} for finding an even number.
• Events with replacement are simpler to calculate compared to "without replacement" because probabilities do not change after each draw.

Understanding this type of event is crucial for problems where the conditions reset after each trial, ensuring consistency in probability calculations.