grabs five of them. Find the probability of the following event, expressing it as a fraction in lowest terms. She has two red ones and one of each of the other colors.
grabs five of them. Find the probability of the following event, expressing it as a fraction in lowest terms. She has two red ones and one of each of the other colors.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Question
![**Problem Statement:**
Whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing three red marbles, two green ones, five white ones, and three purple ones. She grabs five of them. Find the probability of the following event, expressing it as a fraction in lowest terms:
She has two red ones and one of each of the other colors.
**Solution:**
To calculate the probability of this event, we need to determine the number of favorable outcomes and the total number of possible outcomes.
**Favorable Outcomes:**
Suzan needs to grab:
- 2 red marbles out of 3
- 1 green marble out of 2
- 1 white marble out of 5
- 1 purple marble out of 3
Calculate the combinations for each:
- Red: C(3,2) = 3
- Green: C(2,1) = 2
- White: C(5,1) = 5
- Purple: C(3,1) = 3
The number of favorable ways Suzan can grab these specific marbles is:
\[
3 \times 2 \times 5 \times 3 = 90
\]
**Total Possible Outcomes:**
There are a total of 13 marbles (3 red + 2 green + 5 white + 3 purple). Suzan grabs 5 of them, so the total number of ways to choose 5 marbles out of 13 is:
- Total: C(13,5)
Calculate C(13,5):
\[
C(13,5) = \frac{13 \times 12 \times 11 \times 10 \times 9}{5 \times 4 \times 3 \times 2 \times 1} = 1287
\]
**Probability:**
The probability that Suzan grabs two red marbles and one of each of the other colors is the number of favorable outcomes divided by the total number of possible outcomes:
\[
\frac{90}{1287}
\]
**Fraction Simplification:**
Simplify the fraction to its lowest terms.
The final probability is:
\[
\frac{10}{143}
\]
Therefore, the probability that Suzan grabs two red marbles and one of each of the other colors is \(\frac{10}{143}\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb83496cd-9baf-4218-91af-6a9eb71001cb%2F4b256047-83f0-4b8d-bf34-3b044a46e9c3%2Fkedj7p8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing three red marbles, two green ones, five white ones, and three purple ones. She grabs five of them. Find the probability of the following event, expressing it as a fraction in lowest terms:
She has two red ones and one of each of the other colors.
**Solution:**
To calculate the probability of this event, we need to determine the number of favorable outcomes and the total number of possible outcomes.
**Favorable Outcomes:**
Suzan needs to grab:
- 2 red marbles out of 3
- 1 green marble out of 2
- 1 white marble out of 5
- 1 purple marble out of 3
Calculate the combinations for each:
- Red: C(3,2) = 3
- Green: C(2,1) = 2
- White: C(5,1) = 5
- Purple: C(3,1) = 3
The number of favorable ways Suzan can grab these specific marbles is:
\[
3 \times 2 \times 5 \times 3 = 90
\]
**Total Possible Outcomes:**
There are a total of 13 marbles (3 red + 2 green + 5 white + 3 purple). Suzan grabs 5 of them, so the total number of ways to choose 5 marbles out of 13 is:
- Total: C(13,5)
Calculate C(13,5):
\[
C(13,5) = \frac{13 \times 12 \times 11 \times 10 \times 9}{5 \times 4 \times 3 \times 2 \times 1} = 1287
\]
**Probability:**
The probability that Suzan grabs two red marbles and one of each of the other colors is the number of favorable outcomes divided by the total number of possible outcomes:
\[
\frac{90}{1287}
\]
**Fraction Simplification:**
Simplify the fraction to its lowest terms.
The final probability is:
\[
\frac{10}{143}
\]
Therefore, the probability that Suzan grabs two red marbles and one of each of the other colors is \(\frac{10}{143}\
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