Problem 35 - Algebra and Trigonometry

Question

An urn contains 5 white marbles, 10 green marbles, 8 yellow marbles, and 7 black marbles. If one marble is selected, determine the probability that it is white.

Step-by-Step Solution

Answer

The probability that the marble is white is \frac{1}{6}.

Step 1 - Determine the total number of marbles

Add up the number of white, green, yellow, and black marbles to find the total number of marbles in the urn. Total marbles = 5 white + 10 green + 8 yellow + 7 black Calculating this, we get Total marbles = 5 + 10 + 8 + 7 = 30.

Step 2 - Determine the number of favorable outcomes

Identify the number of white marbles, which represent the favorable outcomes, from the urn. The problem states that there are 5 white marbles.

Step 3 - Calculate the probability

Use the probability formula for a single event, which is the number of favorable outcomes divided by the total number of outcomes. The probability that a marble selected is white is given by: \( P(\text{white}) = \frac{\text{Number of white marbles}}{\text{Total number of marbles}} \ P(\text{white}) = \frac{5}{30} \ \ P(\text{white}) = \frac{1}{6} \)

Key Concepts

probability calculations

Probability helps us understand how likely an event is to happen. It's a central concept in statistics and everyday decision-making.
For example, the probability of picking a white marble from an urn can be calculated using a simple formula.
This formula is: \( P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \).
Here, the 'favorable outcomes' are the instances that meet our criteria (a white marble) and the 'total outcomes' are all possible instances (all the marbles).
Understanding this helps us make better predictions and understand randomness in real life.

favorable outcomes

Favorable outcomes are specific results that meet the event criteria.
In our example, picking a white marble is the event, so the favorable outcomes are all the instances where this can occur.
Since the urn contains 5 white marbles, each representing a favorable outcome, we simply count these to determine the number of favorable outcomes.
This counting is essential because it directly impacts the probability calculation.

• 5 white marbles
• 10 green marbles (not favorable)
• 8 yellow marbles (not favorable)
• 7 black marbles (not favorable)

As you can see, only the white marbles are counted towards favorable outcomes.

total outcomes

Total outcomes refer to all possible results that can occur.
In the context of our problem, it means the total number of marbles in the urn.
Adding up all the marbles gives us the total number of outcomes.
From the urn, we have:

• 5 white marbles
• 10 green marbles
• 8 yellow marbles
• 7 black marbles

When we add these together, we get 5 + 10 + 8 + 7 = 30.
So, the total number of possible outcomes is 30.
This number is essential for calculating the probability, as it forms the denominator of our probability formula.
By understanding total outcomes, we ensure we are assessing all possible scenarios.