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Chapter 4: Problem 49

Given that \(A, B\), and \(C\) are three independent events, find their joint probability for the following. a. \(P(A)=.81, \quad P(B)=.49\), and \(P(C)=.36\) b. \(P(A)=.02, \quad P(B)=.03, \quad\) and \(\quad P(C)=.05\)

Short Answer

Expert verified
For the first set of probabilities (a), the joint probability is .14 (approx). For the second set of probabilities (b), the joint probability is .00003.

Step by step solution

01

Understanding Joint Probability

Joint Probability of independent events A, B & C is given as \(P(A \cap B \cap C) = P(A) \cdot P(B) \cdot P(C)\). This is the formula that we'll be using in this exercise.
02

Calculate Joint Probability for the first set of probabilities.

Input the given probabilities into the formula:\(P(A \cap B \cap C) = P(A) \cdot P(B) \cdot P(C) = .81 \cdot .49 \cdot .36\). Now calculate the multiplication to get the result.
03

Calculate Joint Probability for the second set of probabilities.

Input the given probabilities into the formula:\(P(A \cap B \cap C) = P(A) \cdot P(B) \cdot P(C) = .02 \cdot .03 \cdot .05\). Now calculate the multiplication to get the result.

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Key Concepts

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

Understanding Independent Events
Independent events are one of the fundamental concepts in probability. When we say two (or more) events are independent, it means the occurrence of one event does not affect the occurrence of another. Think of it like flipping a coin several times. Each flip is independent, meaning the result of one flip has no effect on the next.
Understanding independence helps simplify the calculation of joint probabilities. It's important to check for independence before applying specific formulas.
  • Independence implies that knowing the outcome of one event provides no information about the other.
  • This concept is crucial when calculating probabilities using specific rules.
Probability Calculation 101
Calculating probabilities is a key skill in statistics. Probabilities express the likelihood of an event occurring. They are values between 0 (impossible event) and 1 (certain event).
For single events, probability is straightforward. However, when dealing with multiple events, it can get more complex if they are not independent.
  • Probability can be expressed as a fraction, decimal, or percentage.
  • In complex scenarios, use rules like the addition and multiplication rules to calculate the probability of combined events.
The calculation method varies based on the relationship between events. But, when the events are independent, the calculations become simpler.
Multiplication of Probabilities for Independent Events
When events are independent, the probability of all occurring together is found by multiplying their individual probabilities. This is known as the multiplication rule of independent events.
The formula is: \[P(A \cap B \cap C) = P(A) \cdot P(B) \cdot P(C)\]This simple rule arises from the fact that each event stands alone, with its own probability unaffected by others. Thus, multiplying these probabilities gives us the joint probability.
  • Ensure the events meet the criteria for independence before using this rule.
  • Apply this rule even when dealing with more than three events, just add more terms to the multiplication.
Remember, this method is streamlined by the independence of the events, making what seems complex remarkably simple.

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Most popular questions from this chapter

What is meant by the joint probability of two or more events? Give one example.

How is the addition rule of probability for two mutually exclusive events different from the rule for two events that are not mutually exclusive?

A player plays a roulette game in a casino by betting on a single number each time. Because the wheel has 38 numbers, the probability that the player will win in a single play is \(1 / 38\). Note that each play of the game is independent of all previous plays. a. Find the probability that the player will win for the first time on the 10 th play. b. Find the probability that it takes the player more than 50 plays to win for the first time. c. A gambler claims that because he has 1 chance in 38 of winning each time he plays, he is certain to win at least once if he plays 38 times. Does this sound reasonable to you? Find the probability that he will win at least once in 38 plays.

A thief has stolen Roger's automatic teller machine (ATM) card. The card has a four-digit personal identification number (PIN). The thief knows that the first two digits are 3 and 5 , but he does not know the last two digits. Thus, the PIN could be any number from 3500 to \(3599 .\) To protect the customer, the automatic teller machine will not allow more than three unsuccessful attempts to enter the PIN. After the third wrong PIN, the machine keeps the card and allows no further attempts. a. What is the probability that the thief will find the correct PIN within three tries? (Assume that the thief will not try the same wrong PIN twice.) b. If the thief knew that the first two digits were 3 and 5 and that the third digit was either 1 or 7 , what is the probability of the thief guessing the correct PIN in three attempts?

An automated teller machine at a local bank is stocked with \(\$ 10\) and \(\$ 20\) bills. When a customer withdraws \(\$ 40\) from the machine, it dispenses either two \(\$ 20\) bills or four \(\$ 10\) bills. If two customers withdraw \(\$ 40\) each, how many outcomes are possible? Draw a tree diagram for this experiment.
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