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Chapter 13: Problem 22

Solve each problem. If a marble is drawn from a bag containing 2 yellow, 5 red, and 3 blue marbles, what is the probability of each event? (a) The marble is yellow or blue. (b) The marble is yellow or red. (c) The marble is green.

Short Answer

Expert verified
a) \( \frac{1}{2} \) b) \( \frac{7}{10} \) c) 0

Step by step solution

01

Determine Total Number of Marbles

First, add up the number of each type of marble to find the total number of marbles in the bag. There are 2 yellow marbles, 5 red marbles, and 3 blue marbles. Therefore, the total number of marbles is: \[ 2 + 5 + 3 = 10 \]
02

Probability of Drawing a Yellow or Blue Marble (Part a)

To find the probability of drawing a yellow or blue marble, add the number of yellow marbles to the number of blue marbles and divide by the total number of marbles. The number of yellow or blue marbles is:\[ 2 + 3 = 5 \]So, the probability is: \[ \frac{5}{10} = \frac{1}{2} \]
03

Probability of Drawing a Yellow or Red Marble (Part b)

To find the probability of drawing a yellow or red marble, add the number of yellow marbles to the number of red marbles and divide by the total number of marbles. The number of yellow or red marbles is:\[ 2 + 5 = 7 \]So, the probability is: \[ \frac{7}{10} \]
04

Probability of Drawing a Green Marble (Part c)

There are no green marbles in the bag. Thus, the number of green marbles is 0. Therefore, the probability is:\[ \frac{0}{10} = 0 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Basic Probability
Probability is the measure of how likely an event is to occur.
It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
For example, if there are 10 marbles in a bag and you want to find the probability of drawing a yellow marble, you look at how many yellow marbles there are.
Since there are 2 yellow marbles out of 10 total marbles, the probability is \(\frac{2}{10} = \frac{1}{5}\).
When working with probability, the results will always be between 0 and 1, where 0 means the event cannot happen, and 1 means the event will always happen.
Dealing with Combined Events
Combined events involve finding the probability of either one event or another happening.
This usually involves the 'or' operator, which means we add the probabilities of the individual events.
For example, if you want to find the probability of drawing a yellow or blue marble from a bag, you would add the probability of drawing a yellow marble to the probability of drawing a blue marble.
In the exercise, there are 2 yellow marbles and 3 blue marbles in a bag of 10 marbles.
The probability of drawing a yellow marble is \(\frac{2}{10} \), and the probability of drawing a blue marble is \(\frac{3}{10} \).
So, the combined probability is \(\frac{2}{10} + \frac{3}{10} = \frac{5}{10} = \frac{1}{2} \).
This tells us there is a 50% chance of drawing either a yellow or blue marble from the bag.
Performing Probability Calculations
To calculate probability, follow these steps:
  • First, determine the total number of possible outcomes. In our exercise, the total number of marbles is 10.
  • Next, identify the number of favorable outcomes for each event. For yellow marbles, there are 2; for blue marbles, there are 3.
  • Calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes. For a yellow or blue marble, this is \(\frac{2+3}{10} = \frac{5}{10} = \frac{1}{2}\).
  • If the problem involves 'or' events (combined events), sum the individual probabilities as shown with yellow or blue marbles.
Always ensure that the probabilities add up correctly to give a logical and valid result.
Understanding basic calculations can make solving more complex probability problems much easier.

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