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Chapter 3: Problem 7

Use the following information to answer the next six exercises. A jar of 150 jelly beans contains 22 red jelly beans, 38 yellow, 20 green, 28 purple, 26 blue, and the rest are orange. Let B = the event of getting a blue jelly bean Let G = the event of getting a green jelly bean. Let O = the event of getting an orange jelly bean. Let P = the event of getting a purple jelly bean. Let R = the event of getting a red jelly bean. Let Y = the event of getting a yellow jelly bean. Find P(G)

Short Answer

Expert verified
The probability of getting a green jelly bean is \( \frac{2}{15} \).

Step by step solution

01

Identify Total Number of Jelly Beans

First, we determine the total number of jelly beans in the jar. According to the problem, there are 150 jelly beans in total.
02

Determine Number of Green Jelly Beans

Next, identify the number of green jelly beans from the information provided. There are 20 green jelly beans in the jar.
03

Calculate Probability of Getting a Green Jelly Bean

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For getting a green jelly bean, the probability is calculated as follows: \[ P(G) = \frac{\text{Number of Green Jelly Beans}}{\text{Total Number of Jelly Beans}} = \frac{20}{150} \]
04

Simplify the Probability

Simplify the fraction obtained to express the probability in the simplest form. \[ P(G) = \frac{20}{150} = \frac{2}{15} \]

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Key Concepts

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

Understanding Favorable Outcomes
In probability, a favorable outcome refers to an event that we are specifically interested in when calculating chance. In our jelly bean exercise, if we want to find the likelihood of picking a green jelly bean, the favorable outcome is exactly this event: getting a green jelly bean.
When working through a probability problem:
  • First, define the event you are interested in - here, it's choosing a green jelly bean.
  • Secondly, determine how many of these events exist 鈥� 20 green jelly beans in this scenario.
Identifying the favorable outcome is the foundation of any probability calculation. It ensures that we know exactly what we're trying to find the probability of, which is crucial for solving the problem correctly.
Calculating Total Outcomes
The total number of outcomes refers to all possible results in the event space. In our jelly bean example, this means the entire set of jelly beans from which we could potentially choose.
There are several steps to determine the total outcomes in a probability question:
  • Add up all the individual groups of elements - the red, yellow, green, purple, blue, and orange jelly beans.
  • Use this sum to calculate the total number, which is 150 jelly beans here.
Understanding this is essential. Without identifying the total number of outcomes, you would not be able to effectively compare the frequency of your event of interest against all the possible events. This comparison is what's needed for the final probability.
Simplifying Fractions in Probability
Once you have identified your favorable outcomes and total outcomes, the next step in calculating probability is often to simplify the fraction that represents these numbers. Simplifying fractions makes them easier to understand and compare.
To simplify a fraction:
  • Find the greatest common divisor of both the numerator (favorable outcomes) and the denominator (total outcomes).
  • Divide both the numerator and the denominator by this greatest common divisor.
In our jelly bean example, the fraction for the probability was initially \(\frac{20}{150}\). By simplifying, we divided both 20 and 150 by their greatest common divisor, which is 10, resulting in \(\frac{2}{15}\). This step is important as it gives us a cleaner, simpler way to express the probability, making it easier to compare with other probabilities.

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Most popular questions from this chapter

Use the following information to answer the next seven exercises. An article in the New England Journal of Medicine, reported about a study of smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans, and 7,650 Whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 Whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 Whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 Whites. Complete the table using the data provided. Suppose that one person from the study is randomly selected. Find the probability that person smoked 11 to 20 cigarettes per day.

Use the following information to answer the next ten exercises. Forty-eight percent of all Californians registered voters prefer life in prison without parole over the death penalty for a person convicted of first degree murder. Among Latino California registered voters, 55% prefer life in prison without parole over the death penalty for a person convicted of first degree murder. 37.6% of all Californians are Latino. In this problem, let: 鈥� C = Californians (registered voters) preferring life in prison without parole over the death penalty for a person convicted of first degree murder. 鈥� L = Latino Californians Suppose that one Californian is randomly selected. Find P(L AND C).

Use the following information to answer the next six exercises. There are 23 countries in North America, 12 countries in South America, 47 countries in Europe, 44 countries in Asia, 54 countries in Africa, and 14 in Oceania (Pacific Ocean region). Let A = the event that a country is in Asia. Let E = the event that a country is in Europe. Let F = the event that a country is in Africa. Let N = the event that a country is in North America. Let O = the event that a country is in Oceania. Let S = the event that a country is in South America. Find P(O).

A shelf holds 12 books. Eight are fiction and the rest are nonfiction. Each is a different book with a unique title. The fiction books are numbered one to eight. The nonfiction books are numbered one to four. Randomly select one book Let F = event that book is fiction Let N = event that book is nonfiction What is the sample space?

Use the following information to answer the next ten exercises. On a baseball team, there are infielders and outfielders. Some players are great hitters, and some players are not great hitters. Let I = the event that a player in an infielder. Let O = the event that a player is an outfielder. Let H = the event that a player is a great hitter. Let N = the event that a player is not a great hitter Write the symbols for the probability that of all the great hitters, a player is an outfielder.
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