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Chapter 7: Problem 60

Sophie is a dog that loves to play catch. Unfortunately, she isn't very good, and the probability that she catches a ball is only \(.1\). Let \(x\) be the number of tosses required until Sophie catches a ball. a. Does \(x\) have a binomial or a geometric distribution? b. What is the probability that it will take exactly two tosses for Sophie to catch a ball? c. What is the probability that more than three tosses will be required?

Short Answer

Expert verified
a. \(x\) has a geometric distribution. b. The probability that it will take exactly two tosses for Sophie to catch a ball is 0.09. c. The probability that more than three tosses will be required is 0.729.

Step by step solution

01

Identifying the Distribution

The distribution can be identified by noting that the exercise is asking for the number of tosses until the first success (Sophie catching the ball). That is the essential characteristic of a geometric distribution.
02

Calculate Probability for Exactly Two Tosses

Using the formula for the geometric distribution \(P(x=k) = (1-p)^{k-1}p\), where p is the probability of success on one trial (0.1 in this case), and k is the number of trials we're interested in (2 in this case), we find that \(P(x=2) = (1-0.1)^{2-1} . 0.1 = 0.9 . 0.1 = 0.09\).
03

Calculate Probability for More than Three Tosses

For this, we use the formula for the cumulative probability of a geometric distribution up to \(k\), represented as \(P(x>k)= (1-p)^k\). Here again, \(p = 0.1\), and \(k = 3\). Therefore, using the formula gives us \(P(x>3) = (1-0.1)^3 = 0.9^3 = 0.729\).

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Key Concepts

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

Probability
Probability is a way to measure how likely an event is to happen. It is expressed as a number between 0 and 1. A probability of 0 means the event will not occur, and a probability of 1 means it is certain to happen.

For example, if you toss a fair coin, there are two possible outcomes: heads or tails. The probability of getting heads is 0.5, and the probability of getting tails is also 0.5. This is because each outcome is equally likely.

Probability helps us to make predictions about future events based on known data. It underlies many concepts in statistics and probability theory such as probability distributions, which describe how probabilities are distributed across outcomes.
Probability Distribution
A probability distribution shows how the probabilities of different possible outcomes are distributed. It gives you a full view of all potential values a random variable can take and how likely each one is to occur.

For discrete random variables, which can take distinct and separate values, a probability distribution tells you the probability of each of these values. For continuous random variables, it describes the probability of the variables falling within a certain range of values.

One of the simplest types is the uniform distribution, where all outcomes are equally likely. On the other hand, a normal distribution, often called a bell curve, is used frequently in statistics to represent real-valued random variables with unknown distributions.

Understanding probability distributions is crucial for analyzing data and making predictions in various fields, such as finance, engineering, and science.
Binomial Distribution
The binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent states and is often used to represent binary outcomes, like success or failure.

It is defined by two parameters: the number of trials (n), and the probability of success (p) on each trial. The probability of getting exactly k successes in n independent Bernoulli trials is given by:

\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

where \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \).

Binomial distributions are used when an experiment involves repeated trials. A key feature is that each trial is independent of the others. This makes it different from a geometric distribution, where we look for the first success in a series of trials.

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Most popular questions from this chapter

Because \(P(z<.44)=.67,67 \%\) of all \(z\) values are less than \(.44\), and \(.44\) is the 67 th percentile of the standard normal distribution. Determine the value of each of the following percentiles for the standard normal distribution (Hint: If the cumulative area that you must look for does not appear in the \(z\) table, use the closest entry): a. The 91 st percentile(Hint: Look for area.9100.) b. The 77 th percentile c. The 50 th percentile d. The 9 th percentile e. What is the relationship between the 70 th \(z\) percentile and the 30 th \(z\) percentile?

Consider a large ferry that can accommodate cars and buses. The toll for cars is \(\$ 3\), and the toll for buses is \(\$ 10\). Let \(x\) and \(y\) denote the number of cars and buses, respectively, carried on a single trip. Cars and buses are accommodated on different levels of the ferry, so the number of buses accommodated on any trip is independent of the number of cars on the trip. Suppose that \(x\) and \(y\) have the following probability distributions: $$ \begin{array}{lrrrrrr} x & 0 & 1 & 2 & 3 & 4 & 5 \\ p(x) & .05 & .10 & .25 & .30 & .20 & .10 \\ y & 0 & 1 & 2 & & & \\ p(y) & .50 & .30 & .20 & & & \end{array} $$ a. Compute the mean and standard deviation of \(x\). b. Compute the mean and standard deviation of \(y\). c. Compute the mean and variance of the total amount of money collected in tolls from cars. d. Compute the mean and variance of the total amount of money collected in tolls from buses. e. Compute the mean and variance of \(z=\) total number of vehicles (cars and buses) on the ferry. f. Compute the mean and variance of \(w=\) total amount of money collected in tolls.

Let \(x\) denote the IQ for an individual selected at random from a certain population. The value of \(x\) must be a whole number. Suppose that the distribution of \(x\) can be approximated by a normal distribution with mean value 100 and standard deviation \(15 .\) Approximate the following probabilities: a. \(P(x=100)\) b. \(\quad P(x \leq 110)\) c. \(P(x<110)\) (Hint: \(x<110\) is the same as \(x \leq 109 .\) ) d. \(\quad P(75 \leq x \leq 125)\)

Airlines sometimes overbook flights. Suppose that for a plane with 100 seats, an airline takes 110 reservations. Define the variable \(x\) as the number of people who actually show up for a sold-out flight. From past experience, the probability distribution of \(x\) is given in the following table: \(\begin{array}{ccccccccc}x & 95 & 96 & 97 & 98 & 99 & 100 & 101 & 102 \\ p(x) & .05 & .10 & .12 & .14 & .24 & .17 & .06 & .04 \\ x & 103 & 104 & 105 & 106 & 107 & 108 & 109 & 110 \\ p(x) & .03 & .02 & .01 & .005 & .005 & .005 & .0037 & .0013\end{array}\) a. What is the probability that the airline can accommodate everyone who shows up for the flight? b. What is the probability that not all passengers can be accommodated? c. If you are trying to get a seat on such a flight and you are number 1 on the standby list, what is the probability that you will be able to take the flight? What if you are number 3 ?

The states of Ohio, Iowa, and Idaho are often confused, probably because the names sound so similar. Each year, the State Tourism Directors of these three states drive to a meeting in one of the state capitals to discuss strategies for attracting tourists to their states so that the states will become better known. The location of the meeting is selected at random from the three state capitals. The shortest highway distance from Boise, Idaho to Columbus, Ohio passes through Des Moines, Iowa. The highway distance from Boise to Des Moines is 1350 miles, and the distance from Des Moines to Columbus is 650 miles. Let \(d_{1}\) represent the driving distance from Columbus to the meeting, with \(d_{2}\) and \(d_{3}\) representing the distances from Des Moines and Boise, respectively. a. Find the probability distribution of \(d_{1}\) and display it in a table. b. What is the expected value of \(d_{1} ?\) c. What is the value of the standard deviation of \(d_{1}\) ? d. Consider the probability distributions of \(d_{2}\) and \(d_{3}\). Is either probability distribution the same as the probability distribution of \(d_{1}\) ? Justify your answer. e. Define a new random variable \(t=d_{1}+d_{2}\). Find the probability distribution of \(t\). f. For each of the following statements, indicate if the statement is true or false and provide statistical evidence to support your answer. i. \(E(t)=E\left(d_{1}\right)+E\left(d_{2}\right)\) (Hint: \(E(t)\) is the expected value of \(t\) and another way of denoting the mean of \(t\).) ii. \(\sigma_{t}^{2}=\sigma_{d_{1}}^{2}+\sigma_{d_{3}}^{2}\)
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