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Chapter 5: Problem 14

Suppose your financial advisor has recommended two stocks, each of which has a 0.6 probability of increasing in value over the next year. Assuming the performance of one stock is independent of the other, what is the probability both stocks will rise over the next year? What is the probability at least one stock will not increase in value?

Short Answer

Expert verified
The probability both stocks will rise is 0.36. The probability at least one stock will not increase is 0.64.

Step by step solution

01

Identify Individual Probabilities

First, identify the probability of each stock increasing in value. Both stocks have a probability of 0.6 of increasing.
02

Calculate Probability That Both Stocks Will Increase

Since the events are independent, multiply the probabilities of each stock increasing. \( P(A \text{ and } B) = P(A) \times P(B) = 0.6 \times 0.6 = 0.36 \)
03

Calculate Probability That At Least One Stock Will Not Increase

First, find the probability that one stock will not increase, which is the complement of the event that it will increase: \( P(\text{not A}) = 1 - 0.6 = 0.4 \). Similarly, \( P(\text{not B}) = 0.4 \). Use the complement rule to find the probability of at least one not increasing: \( P(\text{at least one not increasing}) = 1 - P(\text{both increasing}) = 1 - 0.36 = 0.64 \)

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Key Concepts

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

independent events
In probability theory, two events are considered independent if the occurrence of one does not affect the likelihood of the other. For example, the performance of the two stocks in the exercise is independent.
This means that whether one stock goes up or down has no bearing on whether the other stock will go up or down.
This simplifies the calculations significantly.
When events are independent, we can use the multiplication rule to find the probability of both events occurring together.
complement rule
The complement rule helps us find the probability of the opposite of a certain event.
Simply put, the complement of an event A is the event that A does not happen.
Mathematically, the complement of event A is denoted by \( P(\text{not } A) \).
The rule states that the sum of the probability of an event and its complement is always 1:
\[ P(A) + P(\text{not } A) = 1 \]
Using the exercise, if a stock has a 0.6 probability of increasing:
\( P(\text{not A}) = 1 - P(A) = 1 - 0.6 = 0.4 \).
By understanding this rule, we can find the probability of an event not occurring if we know the probability of it occurring.
multiplication rule
The multiplication rule in probability theory is used to determine the probability of the intersection of two events.
If events A and B are independent, the rule can be expressed as:
\[ P(A \text{ and } B) = P(A) \times P(B) \]
This means we multiply the probabilities of each event occurring.
In the exercise, each stock has a 0.6 probability of increasing in value.
Since both stocks are independent of each other, the probability that both will increase is:
\( P(\text{Both stocks}) = 0.6 \times 0.6 = 0.36 \).
This simple multiplication gives us the joint probability of independent events.

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