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Chapter 7: Problem 66

I t ~ a p p e a r s ~ t h a t ~ t h e r e ~ i s ~ o n l y ~ a ~ o n e ~ i n ~ f i v e ~ c h a n c e ~ t h a t you will be able to take your spring vacation to the Greek Islands. If you are lucky enough to go, you will visit either Corfu ( \(20 \%\) chance) or Rhodes. On Rhodes, there is a \(20 \%\) chance of meeting a tall dark stranger, while on Corfu, there is no such chance.

Short Answer

Expert verified
The probability of meeting a tall dark stranger during your spring vacation is \(16\%\).

Step by step solution

01

Define the events

Let A be the event of going to the Greek Islands, with P(A) = 1/5. Let B be the event of visiting Corfu, with P(B|A) = 20%. Let C be the event of visiting Rhodes, with P(C|A) = 80% (since only Corfu and Rhodes can be visited). Let D be the event of meeting a tall dark stranger.
02

Calculate the probability of meeting a tall dark stranger on Rhodes

Since we are only given the probability of meeting a tall dark stranger while on Rhodes, we have: P(D|C, A) = 20%.
03

Calculate the probability of meeting a tall dark stranger on Corfu

We are told that there is no chance of meeting a tall dark stranger while on Corfu: P(D|B, A) = 0%.
04

Apply the total probability theorem

Now, we are to find the probability of meeting a tall dark stranger during the vacation, or P(D). We will apply the total probability theorem: P(D) = P(D|A) = P(D|B, A)P(B|A) + P(D|C, A)P(C|A)
05

Calculate P(D)

Insert the given probabilities for each of the events: P(D) = (0% 脳 20%) + (20% 脳 80%) P(D) = (0) + (0.2 脳 0.8)
06

Compute the final probability

Calculate the final probability: P(D) = 0.2 脳 0.8 = 0.16 The probability of meeting a tall dark stranger during your spring vacation is 16%.

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Key Concepts

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

Conditional Probability
Understanding conditional probability is essential in resolving problems that involve additional information regarding the occurrence of events. In conditional probability, we focus on finding the probability of an event occurring given that another event has already occurred.
In practical terms, it answers questions such as, "What's the chance of A happening if we know B has occurred?" In this exercise, let's consider:
  • The probability of visiting Corfu given that you are going to the Greek Islands: \( P(B|A) = 0.20 \) or 20%.
  • The probability of meeting a tall dark stranger given that you're specifically on Rhodes: \( P(D|C, A) = 0.20 \) or 20%.
  • No chance of meeting the stranger if you're on Corfu, \( P(D|B, A) = 0 \) or 0%.
Conditional probabilities allow us to zoom in on particular situations, giving our calculations more context and accuracy.
This distinction is especially crucial when different outcomes are possible depending on where you end up during your vacation.
Total Probability Theorem
The total probability theorem is a fundamental concept that helps compute the probability of an event by considering multiple paths to its occurrence. It is particularly useful when an event can happen in several different scenarios or sub-events.
Here's how we apply it in this problem:
  • There are two potential sub-events: going to Corfu or Rhodes.
  • For meeting a tall dark stranger, these sub-events each have different probabilities of occurrence.
The total probability of meeting a tall dark stranger during the vacation, \( P(D) \), is calculated by summing the probabilities of meeting a stranger on either Corfu or Rhodes:
When you break it down, you follow this formula:\[ P(D) = P(D|B, A)P(B|A) + P(D|C, A)P(C|A) \]
This sum takes into account all potential paths to meet the stranger, weighted according to their likelihood.
Event Outcomes
Event outcomes refer to all possible results that could occur in a given situation. Each outcome has a probability, and the sum of all probabilities equals 1. Understanding all potential outcomes gives us a complete picture of how events are likely to unfold.
In the exercise, there are specific outcomes:
  • Going to the Greek Islands, which has a 20% chance of directing you to Corfu.
  • An 80% chance of ending up on Rhodes.
  • A 16% overall chance of meeting a tall dark stranger.
Each of these percentages represents distinct possibilities. By identifying and differentiating between "meeting a stranger on Rhodes" and "not meeting one on Corfu," we consider the unique outcomes possible from this vacation adventure.
By carefully analyzing these potential outcomes and their probabilities, we gain insights into the probable scenarios that could come to pass.

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Most popular questions from this chapter

\(\nabla\) Two distinguishable dice are rolled. Could there be two mutually exclusive events that both contain outcomes in which the numbers facing up add to 7 ?

Let \(E\) be the event that you meet a tall dark stranger. Which of the following could reasonably represent the experiment and sample space in question? (A) You go on vacation and lie in the sun; \(S\) is the set of cloudy days. (B) You go on vacation and spend an evening at the local dance club; \(S\) is the set of people you meet. (C) You go on vacation and spend an evening at the local dance club; \(S\) is the set of people you do not meet.

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Complete the following probability distribution table, and then calculate the stated probabilities. $$ \begin{array}{|r|c|c|c|c|c|} \hline \text { Outcome } & \mathrm{a} & \mathrm{b} & \mathrm{c} & \mathrm{d} & \mathrm{e} \\ \hline \text { Probability } & .1 & .05 & .6 & .05 & \\ \hline \end{array} $$ a. \(P(\\{a, c, e)\\}\) b. \(P(E \cup F),\) where \(E=\\{\mathrm{a}, \mathrm{c}, \mathrm{e}\\}\) and \(F=\\{\mathrm{b}, \mathrm{c}, \mathrm{e}\\}\) c. \(P\left(E^{\prime}\right),\) where \(E\) is as in part (b) d. \(P(E \cap F)\), where \(E\) and \(F\) are as in part (b)

Use counting arguments from the preceding chapter. Horse Races The seven contenders in the fifth horse race at Aqueduct on February 18,2002, were: Pipe Bomb, Expect a Ship, All That Magic, Electoral College, Celera, Cliff Glider, and Inca Halo. \({ }^{5}\) You are interested in the first three places (winner, second place, and third place) for the race. a. Find the cardinality \(n(S)\) of the sample space \(S\) of all possible finishes of the race. (A finish for the race consists of a first, second, and third place winner.) b. Let \(E\) be the event that Electoral College is in second or third place, and let \(F\) be the event that Celera is the winner. Express the event \(E \cap F\) in words, and find its cardinality.
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