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Chapter 1: Problem 12

Let \(E\) and \(F\) be mutually exclusive events in the sample space of an experiment. Suppose that the experiment is repeated until either event \(E\) or event \(F\) occurs. What does the sample space of this new super experiment look like? Show that the probability that event \(E\) occurs before event \(F\) is \(P(E) /\) \([P(E)+P(F)]\) Hint: Argue that the probability that the original experiment is performed \(n\) times and \(E\) appears on the \(n\) th time is \(P(E) \times(1-p)^{n-1}, n=1,2, \ldots\), where \(p=P(E)+P(F)\). Add these probabilities to get the desired answer.

Short Answer

Expert verified
The probability that event \(E\) occurs before event \(F\) in the given experiment is \(P(E\text{ before }F) = \frac{P(E)}{P(E) + P(F)}\).

Step by step solution

01

Identify Mutually Exclusive Events

Mutually exclusive events are events that cannot happen at the same time. In other words, if one of the events happens, the other event cannot happen. In this case, events \(E\) and \(F\) cannot both happen simultaneously. That is, \(P(E \cap F) = 0\).
02

Understand Repeated Experiment

The experiment is repeated until either event \(E\) or event \(F\) occurs. This means that we stop the experiment once we have observed either \(E\) or \(F\), whichever comes first.
03

Define Sample Space and Probability

The sample space of the new super experiment can be represented as sequences of trials until either event \(E\) or event \(F\) occurs, such as: \(E, FE, FFE, FFFE, ...\) Let the probability that the original experiment is performed \(n\) times, and \(E\) appears on the \(n\) th time be: \(P_n(E) = P(E) \times (1-p)^{n-1}, n=1,2, \ldots\), where \(p = P(E) + P(F)\).
04

Determine Probability That E Occurs Before F

To find the probability that event \(E\) occurs before event \(F\), we need to add the probabilities of event \(E\) occurring on the \(n\)th time for all possible \(n\). Using the formula in Step 3, the total probability is given by: \(P(E\text{ before }F) = \sum_{n=1}^{\infty} P_n(E) = \sum_{n=1}^{\infty} \left[P(E) \times (1-p)^{n-1}\right]\).
05

Calculate the Probability

To find the sum of the infinite series, we can use the formula for the sum of an infinite geometric series: \(\sum_{n=1}^{\infty} ar^{n-1} = \frac{a}{1-r}\), where \(a\) is the first term in the series, and \(r\) is the common ratio. In this case, the sum can be written as: \(P(E\text{ before }F) = \frac{P(E)}{1-(1-p)}\), and since \(1-(1-p) = p\), we get \(P(E\text{ before }F) = \frac{P(E)}{P(E) + P(F)}\). Hence, the probability that event \(E\) occurs before event \(F\) is: \(P(E\text{ before }F) = \frac{P(E)}{P(E) + P(F)}\).

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Key Concepts

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

Mutually Exclusive Events
Understanding mutually exclusive events is essential when studying probability. These are events that cannot occur at the same time, meaning the occurrence of one event excludes the possibility of the other event happening. In mathematical terms, when events E and F are mutually exclusive, the probability that both E and F happen together is zero, represented as \( P(E \cap F) = 0 \).

Here鈥檚 an example to illustrate this concept: suppose you have a deck of cards, and event E is drawing a heart, while event F is drawing a club. You cannot draw a card that is both a heart and a club at the same time, hence, these are mutually exclusive events. This provides a straightforward approach to calculating probabilities in scenarios where any one of several outcomes can occur, but never simultaneously.
Repeated Experiments
In probability, a repeated experiment refers to performing the same trial over and over until a specific outcome is achieved. The outcomes of each trial are independent of one another, and the probability remains constant across trials. This is typically visualized using a tree diagram or sequences illustrating different possibilities for each repeat.

In the given exercise, the experiment is performed repeatedly until we observe either event E or F. It's like flipping a coin until it lands heads (E) or tails (F) 鈥� we repeat the flipping, each time with an equal chance for heads or tails, until we see one of them. Understanding the nature of repeated experiments helps with predicting outcomes over multiple tries and calculating the overall probability of an event occurring.
Infinite Geometric Series
An infinite geometric series is a sequence of numbers with a common ratio between consecutive terms, which goes on indefinitely. It can be summed up under certain conditions, specifically when the absolute value of the common ratio is less than one. The formula for the sum of an infinite geometric series is given by \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio.

In the context of probabilities, when dealing with repeated experiments, we often calculate the sum of the probabilities across an infinite number of trials, as seen in our original problem. By recognizing the structure of an infinite geometric series in the probability calculations, we can find the sum and thus the overall probability of an event occurring over an indefinite number of tries. This concept is pivotal when evaluating long-term behaviors and expectations in various probabilistic scenarios.

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Most popular questions from this chapter

An individual uses the following gambling system at Las Vegas. He bets \(\$ 1\) that the roulette wheel will come up red. If he wins, he quits. If he loses then he makes the same bet a second time only this time he bets \(\$ 2\); and then regardless of the outcome, quits. Assuming that he has a probability of \(\frac{1}{2}\) of winning each bet, what is the probability that he goes home a winner? Why is this system not used by everyone?

Stores \(A, B\), and \(C\) have 50,75, and 100 employees, and, respectively, 50, 60 , and 70 percent of these are women. Resignations are equally likely among all employees, regardless of sex. One employee resigns and this is a woman. What is the probability that she works in store \(C\) ?

Bill and George go target shooting together. Both shoot at a target at the same time. Suppose Bill hits the target with probability \(0.7\), whereas George, independently, hits the target with probability \(0.4 .\) (a) Given that exactly one shot hit the target, what is the probability that it was George's shot? (b) Given that the target is hit, what is the probability that George hit it?

Suppose all \(n\) men at a party throw their hats in the center of the room. Each man then randomly selects a hat. Show that the probability that none of the \(n\) men selects his own hat is $$ \frac{1}{2 !}-\frac{1}{3 !}+\frac{1}{4 !}-+\cdots \frac{(-1)^{n}}{n !} $$ Note that as \(n \rightarrow \infty\) this converges to \(e^{-1}\). Is this surprising?

In a certain species of rats, black dominates over brown. Suppose that a black rat with two black parents has a brown sibling. (a) What is the probability that this rat is a pure black rat (as opposed to being a hybrid with one black and one brown gene)? (b) Suppose that when the black rat is mated with a brown rat, all five of their offspring are black. Now, what is the probability that the rat is a pure black rat?
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