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Chapter 4: Problem 89

The probability that a farmer is in debt is 80 . What is the probability that three randomly selected farmers are all in debt? Assume independence of events.

Short Answer

Expert verified
The probability that three randomly selected farmers are all in debt is 51.2%.

Step by step solution

01

Understand the independence of events

Understand that if the events are independent, the probability of all the events happening is calculated by multiplying the probability of each event. In this case, the event is a farmer being in debt, and its probability is 0.80.
02

Multiply the probabilities

Since we are finding the probabilities of three farmers all being in debt, and the events are independent, we need to multiply the probability of one farmer being in debt thrice (\(0.80 * 0.80 * 0.80\) or \(0.80^3\)).
03

Calculate the probability

Upon multiplying, we get \(0.80 * 0.80 * 0.80 = 0.512\) or 51.2%. This means, there is a 51.2% probability that three randomly selected farmers are all in debt assuming independence of events.

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Key Concepts

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

Independent Events
In probability, understanding whether events are independent is crucial. Independent events mean that the occurrence of one event does not affect the occurrence of another. For example, if one farmer being in debt doesn鈥檛 influence another farmer's financial situation, these events are considered independent.

This is an important assumption in probability problems that involve multiple trials or selections. It simplifies calculations because it allows us to treat each event as separate, even though they occur back-to-back or together.
Multiplication Rule
The multiplication rule in probability is used when we want to find the probability of two or more independent events all occurring. It's simple: you multiply the probabilities of each individual event.
  • If you have event A and event B, both independent, the probability of both A and B occurring is \( P(A \text{ and } B) = P(A) \times P(B) \).
This rule tells us how to combine probabilities when events are not influencing each other. In our farmer problem, we used this rule to find the probability of all three farmers being in debt by multiplying the probability for one farmer being in debt three times.
Probability Calculation
Calculating probabilities might seem complex, but it鈥檚 just about understanding and applying simple rules. To find the probability of a series of independent events happening together, we use the multiplication rule.
  • Let鈥檚 say the probability of a single event (a farmer in debt) is 0.80.
  • For three independent farmers being in debt, we multiply \(0.80 \times 0.80 \times 0.80 = 0.512\).
Thus, the probability of three farmers all being in debt is 51.2%. Breaking it down into smaller steps helps ensure that calculations are done correctly and confusion is minimized.
Random Selection
Random selection means picking items in a way that each item has an equal chance of being chosen. It's essential in probability to ensure fairness and lack of bias.

In our scenario, randomly selecting farmers ensures that there's no systematic influence on whether a particular farmer is in debt. This randomness allows us to apply probability rules accurately. When events are independently selected at random, each selection is free from the other selections, supporting the notion of independent events.

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Most popular questions from this chapter

A random sample of 400 college students was asked if college athletes should be paid. The following table gives a two-way classification of the responses. $$ \begin{array}{lcc} \hline & \text { Should Be Paid } & \text { Should Not Be Paid } \\ \hline \text { Student athlete } & 90 & 10 \\ \text { Student nonathlete } & 210 & 90 \\ \hline \end{array} $$ a. If one student is randomly selected from these 400 students, find the probability that this student i. is in favor of paying college athletes ii. favors paying college athletes given that the student selected is a nonathlete iii. is an athlete and favors paying student athletes iv. is a nonathlete \(o r\) is against paying student athletes b. Are the events "student athlete" and "should be paid" independent? Are they mutually exclusive? Explain why or why not.

The Big Six Wheel (or Wheel of Fortune) is a casino and carnival game that is well known for being a big money maker for the casinos. The wheel has 54 equally likely slots (outcomes) on it. The slot that pays the largest amount of money is called the joker. If a player bets on the joker, the probability of winning is \(1 / 54\). The outcome of any given play of this game (a spin of the wheel) is independent of the outcomes of previous plays. a. Find the probability that a player who always bets on joker wins for the first time on the 15 th play of the game. b. Find the probability that it takes a player who always bets on joker more than 70 plays to win for the first time.

A pizza parlor has 12 different toppings available for its pizzas, and 2 of these toppings are pepperoni and anchovies. If a customer picks 2 toppings at random, find the probability that a. neither topping is anchovies b. pepperoni is one of the toppings

In a statistics class of 42 students, 28 have volunteered for community service in the past. If two students are selected at random from this class, what is the probability that both of them have volunteered for community service in the past? Draw a tree diagram for this problem.

The probability that a corporation makes charitable contributions is .72. Two corporations are selected at random, and it is noted whether or not they make charitable contributions. a. Draw a tree diagram for this experiment. b. Find the probability that at most one corporation makes charitable contributions.
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