A spinner with 8 equal sections was spun 140 times. What is a reasonable prediction for the number of times the spinner will land on an even number? step by step explanation

To predict the number of times the spinner will land on an even number out of 140 spins, we can follow these steps:

Step 1: Understand the Spinner Our spinner has 8 equal sections. We need to figure out how many of those sections represent even numbers. Typically, a spinner with numbers will have a consecutive sequence of numbers. Assuming this is true, there will be 4 even numbers (2, 4, 6, and 8) and 4 odd numbers (1, 3, 5, and 7).

Step 2: Calculate the Probability We want to calculate the probability of the spinner landing on an even number on any given spin. Since there are 4 even numbers out of 8, the probability (P) of landing on an even number is P(even) = 4/8, which simplifies to P(even) = 1/2 or 0.5.

Step 3: Apply the Probability Now that we know the probability of the spinner landing on an even number is 0.5, we can apply it to the total number of spins (140) to make a prediction. The expected value (E) is found by multiplying the probability by the total number of trials: E(even) = P(even) * Total Spins.

Step 4: Calculation E(even) = 0.5 * 140 = 70.

Step 5: Making the Prediction Based on our calculations, a reasonable prediction for the number of times the spinner will land on an even number is 70 times out of 140 spins.

This prediction assumes that the spinner is fair and every section has an equal chance of being landed on. In practice, the actual result may vary due to chance and potential imperfections in the spinner, but 70 is a reasonable expectation based on probability.

Extra: Understanding Probability: Probability is the branch of mathematics that deals with the likelihood of various outcomes. It's often expressed as a number between 0 and 1, or as a percentage. A probability of 0 means an event will not occur, and a probability of 1 means it will certainly occur. In this scenario, the probability of landing on an even number, 0.5, implies that there is a 50% chance at each spin.

Calculating Expected Value: The expected value is a statistical concept that provides the average outcome one might expect after a large number of trials. It is found by multiplying the probability of each outcome by the value of each outcome and adding these products together. In the context of the spinner, the expected value tells us the average number of times the spinner should land on an even number after many spins.

Predictions vs. Reality: While probability and expected values offer a way to make predictions about random events, it's important to note that the outcomes of actual trials can differ from predictions. This is due to variability and randomness inherent in any process. However, as the number of trials increases, the actual results should come closer to the expected outcomes due to the law of large numbers, which states that as an experiment is repeated a large number of times, the average of the results should be close to the expected value.