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 black.

Short answer

The probability is \( \frac{7}{30} \).

Step-by-step solution

- Determine the Total Number of Marbles

Add up all the marbles in the urn. There are 5 white, 10 green, 8 yellow, and 7 black marbles.Total number of marbles = 5 + 10 + 8 + 7

- Calculate the Total

Perform the addition to find the total number of marbles.Total number of marbles = 5 + 10 + 8 + 7 = 30

- Identify the Number of Black Marbles

Find the number of black marbles given in the problem, which is 7.

- Use the Probability Formula

The probability of an event is calculated using the formula: \[ P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]In this case, the event is selecting a black marble.

- Compute the Probability

Substitute the values into the probability formula. The number of favorable outcomes (selecting a black marble) is 7, and the total number of outcomes (total marbles) is 30.\[ P(\text{Black}) = \frac{7}{30} \]

Key concepts

probability formula

Probability is a way to measure how likely an event is to happen. We use the probability formula to calculate this likelihood. The formula is:

\[ P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]

The formula might look complex at first. But it simply means you need to divide the number of ways an event can happen (favorable outcomes) by the total number of possible outcomes.

For example, if you want to find the probability of drawing a black marble from our urn, you need to know two things:

• The number of black marbles (favorable outcomes).
• The total number of marbles in the urn (total outcomes).

favorable outcomes

To calculate probability, you need to identify the favorable outcomes. Let's break down what favorable outcomes mean. Favorable outcomes are the specific outcomes you are interested in.

In the context of the given exercise, we are looking at the probability of drawing a black marble. So, the favorable outcomes are the number of black marbles in the urn.

According to the problem, there are 7 black marbles in the urn.

Always clearly identify what you consider a 'favorable outcome' before using the probability formula.

total outcomes

Total outcomes refer to all the possible outcomes that can occur in a situation. For our probability exercise, we need to consider every marble present in the urn.

Here, the urn contains:

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

To find the total number of outcomes, you sum all these marbles:

\[ \text{Total outcomes} = 5 + 10 + 8 + 7 = 30 \]

So, in our problem, there are 30 marbles total in the urn.

marbles probability exercise

Now that we understand both favorable outcomes and total outcomes, let’s solve the exercise. We want to compute the probability of drawing a black marble:

Step-by-step, here's what we do:

• Add up all the marbles in the urn: \[ 5 + 10 + 8 + 7 = 30 \].
• Identify the number of black marbles (favorable outcomes): 7.
• Apply the probability formula: \[ P(\text{Black}) = \frac{7}{30} \].

The probability of selecting a black marble from the urn is \[ \frac{7}{30} \].

And there you have it! Understanding these steps and concepts helps demystify probability and makes solving similar problems straightforward.