Problem 31 - Algebra 2

Question

Determine whether the events are independent or dependent. Then find the probability. A cage contains 3 white and 6 brown hamsters. Maggie randomly selects one, puts it back, and then randomly selects another. What is the probability that both selections were white?

Step-by-Step Solution

Answer

The events are independent, and the probability of both selections being white is \( \frac{1}{9} \).

Step 1: Understand the Event

First, determine if the events are independent or dependent. When an event is independent, the outcome of that event does not affect the outcome of another event. Since Maggie puts the hamster back each time before selecting again, the selections are independent.

Step 2: Calculate Probability of First Event

Calculate the probability that the first hamster selected is white. There are 3 white hamsters and 9 total hamsters (3 white + 6 brown), so the probability of selecting a white hamster first is \( \frac{3}{9} = \frac{1}{3} \).

Step 3: Calculate Probability of Second Event

Since the first hamster is put back, the probability of selecting a white hamster again remains the same. The probability of selecting a white hamster in the second draw is also \( \frac{3}{9} = \frac{1}{3} \).

Step 4: Find Combined Probability

Since both events are independent, multiply the probabilities of each event to find the combined probability: \( \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} \). Thus, the probability that both selections are white is \( \frac{1}{9} \).

Key Concepts

Probability

Probability is a mathematical concept that measures the likelihood of something happening. It is often expressed as a fraction between 0 and 1. A probability of 0 means an event will not occur, while a probability of 1 means the event is certain to occur.

To calculate probability, consider the ratio of the number of favorable outcomes to the total number of possible outcomes. For instance, if you want to know the probability of drawing a white hamster from a group of 9, you count that there are 3 white hamsters and 9 total hamsters, resulting in a probability of \( \frac{3}{9} \), which simplifies to \( \frac{1}{3} \). Calculating probability often involves basic arithmetic that helps us predict how likely an event is to happen.

Independent Versus Dependent Events

Understanding whether events are independent or dependent is crucial in calculating probabilities. Independent events mean the outcome of one event does not influence the outcome of another. In the hamster selection example, since Maggie puts each hamster back before the next selection, each event is independent.

Dependent events, on the other hand, occur when the outcome of one event affects the outcome of another. For example, if Maggie did not return the hamster to the cage after the first selection, the second event would be dependent because the total number of hamsters changes.

When events are independent, the occurrence of one does not change the probability of the other. This distinction is key in correctly using probability rules to calculate accurate probabilities.

Probability Multiplication Rule

The probability multiplication rule is used to determine the combined probability of two or more independent events. For independent events, you simply multiply the individual probabilities.

In the case of Maggie selecting a white hamster, the probability of picking a white hamster in one draw is \( \frac{1}{3} \). Since each draw is independent, the probability that both selections are white is \( \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} \).

This rule helps in understanding scenarios where multiple independent actions occur, allowing for the calculation of the likelihood of both actions happening according to the individual probabilities designated for each event.