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Chapter 4: Problem 70

Find the joint probability of \(A\) and \(B\) for the following. a. \(P(A)=.40\) and \(P(B \mid A)=.25\) b. \(P(B)=.65\) and \(P(A \mid B)=.36\)

Short Answer

Expert verified
a. The joint probability of A and B is 10%.\nb. The joint probability of A and B is 23.4%.

Step by step solution

01

Compute joint probability for part (a)

Use the formula \(P(A \cap B) = P(A) \cdot P(B|A)\) to find the joint probability of events A and B. Here, \(P(A)\) is 0.40 and \(P(B|A)\) is 0.25. Thus, the joint probability \(P(A \cap B) =0.40 \cdot 0.25=0.10=10%\).
02

Compute joint probability for part (b)

Use the formula \(P(A \cap B) = P(B) \cdot P(A|B)\) to find the joint probability of events A and B. Here, \(P(B)\) is 0.65 and \(P(A|B)\) is 0.36. Thus, the joint probability \(P(A \cap B) =0.65 \cdot 0.36 =0.234=23.4%\).

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Key Concepts

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

Conditional Probability
Conditional probability is all about finding the likelihood of an event happening given that another event has already occurred. Imagine you're already enjoying a sunny day, and you want to know the chances of also seeing a rainbow. This is where conditional probability comes in handy.

Here鈥檚 how it works:
  • The notation \(P(B|A)\) represents the probability of event \(B\) given that \(A\) has occurred.
  • It's calculated using data or assumptions and helps provide more context-specific probabilities.
  • In the exercise, \(P(B|A) = 0.25\) means there鈥檚 a 25% chance of \(B\) happening if \(A\) has already happened.
With conditional probabilities, we better understand how one event can influence another. Always keep in mind: it's essential to know the conditions since they can change the probability drastically.
Probability Theory
Probability theory is the mathematics of uncertainty. It helps us model situations where outcomes are uncertain and provides tools to determine how likely these outcomes are.

Key concepts include:
  • Events: These are the outcomes or results we are interested in. For example, rolling a dice and getting a number 6.
  • Probabilities: These are numbers between 0 and 1 that describe how likely an event is. A probability of 0 means the event will not happen, whereas 1 means the event is certain.
In our problem:
  • \(P(A) = 0.40\) tells us there's a 40% chance of event \(A\) occurring.
  • Similarly, \(P(B) = 0.65\) indicates event \(B\) has a 65% likelihood.
Using probability theory, we explore the interactions between events, like determining joint probabilities, to understand how often two or more events happen together.
Probability Formulas
The formulas of probability are the computational rules that help us find these chances mathematically. For joint probability, which is the probability that both events \(A\) and \(B\) happen, we use specific formulas.

For joint probability:
  • "And" means both events occur, represented as \(P(A \cap B)\).
  • In the exercise, the formula for joint probability is \(P(A \cap B) = P(A) \cdot P(B|A)\) or \(P(B) \cdot P(A|B)\).
  • It makes use of conditional probabilities to find the likelihood of two events coinciding.
Here's how they were applied:
  • In part (a), \(P(A \cap B)\) was calculated using \(P(A)\) and \(P(B|A)\) as \(0.40 \times 0.25 = 0.10\), meaning a 10% chance both \(A\) and \(B\) occur.
  • In part (b), \(0.65 \times 0.36 = 0.234\), indicating a 23.4% likelihood for both \(A\) and \(B\).
The mastery of these formulas provides a systematic way to approach problems involving multiple probabilities and their interactions.

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Most popular questions from this chapter

A random sample of 80 lawyers was taken, and they were asked if they are in favor of or against capital punishment. The following table gives the two-way classification of their responses. $$\begin{array}{lcc} \hline & \begin{array}{c} \text { Favors Capital } \\ \text { Punishment } \end{array} & \begin{array}{c} \text { Opposes Capital } \\ \text { Punishment } \end{array} \\ \hline \text { Male } & 32 & 24 \\ \text { Female } & 13 & 11 \\ \hline \end{array}4\( a. If one lawyer is randomly selected from this group, find the probability that this lawyer i. favors capital punishment ii. is a female iii. opposes capital punishment given that the lawyer is a female iv. is a male given that he favors capital punishment \)\mathrm{v}\(. is a female and favors capital punishment vi. opposes capital punishment \)o r$ is a male b. Are the events "female" and "opposes capital punishment" independent? Are they mutually exclusive? Explain why or why not.

Given that \(P(B \mid A)=.80\) and \(P(A\) and \(B)=.58\), find \(P(A)\).

According to a 2007 America's Families and Living Arrangements Census Bureau survey, \(52.1\) million children lived with both of their parents in the same household, whereas \(21.6\) million lived with at most one parent in the household. Assume that all U.S. children are included in this survey and that this information is true for the current population. If one child is selected at random, what are the two complementary events and their probabilities?

When is the following addition rule used to find the probability of the union of two events \(A\) and \(B\) ? $$P(A \text { or } B)=P(A)+P(B)$$ Give one example where you might use this formula.

A man just bought 4 suits, 8 shirts, and 12 ties. All of these suits, shirts, and ties coordinate with each other. If he is to randomly select one suit, one shirt, and one tie to wear on a certain day, how many different outcomes (selections) are possible?
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