Find the probability that an investor makes a profit on 6 of her next 12 investments, assuming independence and a 0.6 chance of making a profit each time.
Find the probability that an investor makes a profit on 6 of her next 12 investments, assuming independence and a 0.6 chance of making a profit each time.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Probability Calculation for Independent Investments**
*Problem Statement:*
**8**) Find the probability that an investor makes a profit on 6 of her next 12 investments, assuming independence and a 0.6 chance of making a profit each time.
**Explanation:**
To find this probability, we can use the binomial probability formula, given that each investment is independent and has the same probability of profit (0.6).
The binomial probability formula is:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
Where:
- \( P(X = k) \) is the probability of making a profit on \( k \) investments out of \( n \) total investments.
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \).
- \( p \) is the probability of making a profit on a single investment.
- \( n \) is the total number of investments.
- \( k \) is the desired number of successful investments (profits).
For this problem:
- \( n = 12 \)
- \( k = 6 \)
- \( p = 0.6 \)
Substitute these values into the binomial probability formula to find the desired probability.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F464fe16a-00d5-4182-8379-61bd4b8bbe02%2Fb071a872-f254-417b-9657-d0e673557af6%2F7khjt4_processed.png&w=3840&q=75)
Transcribed Image Text:**Probability Calculation for Independent Investments**
*Problem Statement:*
**8**) Find the probability that an investor makes a profit on 6 of her next 12 investments, assuming independence and a 0.6 chance of making a profit each time.
**Explanation:**
To find this probability, we can use the binomial probability formula, given that each investment is independent and has the same probability of profit (0.6).
The binomial probability formula is:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
Where:
- \( P(X = k) \) is the probability of making a profit on \( k \) investments out of \( n \) total investments.
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \).
- \( p \) is the probability of making a profit on a single investment.
- \( n \) is the total number of investments.
- \( k \) is the desired number of successful investments (profits).
For this problem:
- \( n = 12 \)
- \( k = 6 \)
- \( p = 0.6 \)
Substitute these values into the binomial probability formula to find the desired probability.
Transcribed Image Text:### Problem-Solving Steps for Probability Distributions
For each problem:
a) **Identify the Best Distribution**
- Determine whether the best distribution to solve the problem is **Binomial**, **Poisson**, or **Exponential**.
b) **Identify Parameters**
- Identify the correct value of all associated parameters for the chosen distribution.
c) **Calculate the Probability**
- Find the indicated probability and round it to 4 decimal places.
---
#### Explanation of Probability Distributions:
1. **Binomial Distribution**
- Used when there are a fixed number of trials, each with two possible outcomes (success/failure), and the probability of success is the same in each trial.
- Parameters: \( n \) (number of trials) and \( p \) (probability of success).
2. **Poisson Distribution**
- Used for counting the number of events that occur in a fixed interval of time or space, where the events occur with a known constant mean rate and independently of the time since the last event.
- Parameter: \( \lambda \) (average rate of occurrence).
3. **Exponential Distribution**
- Used to model the time between events in a Poisson process, i.e., for situations where an event occurs continuously and independently at a constant average rate.
- Parameter: \( \lambda \) (rate parameter, which is the reciprocal of the mean).
Follow these steps systematically to solve probability problems effectively using the appropriate distribution.
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