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Chapter 5: Problem 94

Late flights An airline reports that 85% of its flights arrive on time. To find the probability that its next four flights into LaGuardia Airport all arrive on time, can we multiply (0.85)(0.85)(0.85)(0.85)? Why or why not?

Short Answer

Expert verified
Yes, assuming flights are independent, multiplying 0.85 four times gives the probability of all arriving on time as 52.2%.

Step by step solution

01

Define Probability of an Event

The probability of an event is a measure of the likelihood that the event will occur. Here, the probability of one flight arriving on time is 0.85 or 85%.
02

Understand the Context

The question asks whether the next four flights arriving on time can be calculated simply by multiplying the probability of one flight arriving on time four times. This assumes that the events are independent.
03

Check for Independence

In probability, events are independent if the occurrence of one does not affect the probability of the other occurring. We need to analyze if each flight's arrival is independent of the others. For this exercise, we assume flights arriving on time are independent, meaning no flight's arrival is influenced by another.
04

Calculate Probability of All Independent Events

If the flights are independent, the probability of all four flights arriving on time is the product of their individual probabilities. Therefore, we calculate this as: \( P = (0.85)^4 \).
05

Calculation

Calculate \( (0.85)^4 \):\[ (0.85)^4 = 0.85 \times 0.85 \times 0.85 \times 0.85 = 0.52200625 \]Thus, the probability of all four flights arriving on time is approximately 0.522 or 52.2%.

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Key Concepts

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

Independence of Events
Independence in probability means that the outcome of one event does not affect the outcome of another. Imagine flipping a coin: the result of one flip does not influence the next. This is a classic example of independent events because each flip is separate.
In the context of the airline flights, we assume that whether one flight arrives late or on time does not affect the others. This means that each flight's timeliness is not influenced by delays or punctuality in previous flights.
  • If flights are independent, the scenario allows for straightforward calculations as each flight's on-time probability remains constant regardless of other flights.
  • This assumption simplifies mathematical modeling and helps in predicting the likelihood of multiple events occurring together.
Probability Calculation
Calculating the probability of an event involves determining how likely that event is to happen among all possible outcomes. Probability ranges from 0 (impossible event) to 1 (certain event).
In our problem, the probability of one airline flight arriving on time at LaGuardia Airport is 0.85 or 85%. This means that out of all flights, 85 out of 100 are expected to be on time.
  • Single-event probability gives a snapshot: knowing one flight's on-time probability helps assess future occurrences.
  • The focus here is on individual flights, using this base probability to explore more complex scenarios.
Multiplication Rule of Probability
The Multiplication Rule is a crucial tool in calculating the probability of several independent events occurring in sequence. According to this rule, the overall probability is found by multiplying the probabilities of each individual event.
For example, if you want to find the probability of four independent flights all arriving on time, you would compute this as the product of each flight's independent probabilities: \[ P = 0.85 \times 0.85 \times 0.85 \times 0.85 \]
  • This rule applies only when events are independent, ensuring each event's occurrence does not influence others.
  • Using the multiplier approach simplifies complex probability calculation into manageable steps.
The calculation yields \( P = (0.85)^4 = 0.52200625 \), which indicates a 52.2% probability for all four flights to be on time.

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Most popular questions from this chapter

An unenlightened gambler (a) A gambler knows that red and black are equally likely to occur on each spin of a roulette wheel. He observes five consecutive reds occur and bets heavily on black at the next spin. Asked why, he explains that black is due by the law of averages. Explain to the gambler what is wrong with this reasoning. (b) After hearing you explain why red and black are still equally likely after five reds on the roulette wheel, the gambler moves to a poker game. He is dealt five straight red cards. He remembers what you said and assumes that the next card dealt in the same hand is equally likely to be red or black. Is the gambler right or wrong, and why?

The casino game craps is based on rolling two dice. Here is the assignment of probabilities to the sum of the numbers on the up-faces when two dice are rolled: The most common bet in craps is the 鈥減ass line.鈥� A pass line bettor wins immediately if either a 7 or an 11 comes up on the first roll. This is called a natural. What is the probability of a natural? $$ \begin{array}{ll}{\text { (a) } 2 / 36} & {\text { (c) } 8 / 36 \quad \text { (e) } 20 / 36} \\ {\text { (b) } 6 / 36} & {\text { (d) } 12 / 36}\end{array} $$

Probability models? In each of the following situations, state whether or not the given assignment of probabilities to individual outcomes is legitimate, that is, satisfies the rules of probability. If not, give specific reasons for your answer. (a) Roll a die and record the count of spots on the up-face: \(P(1)=0, P(2)=1 / 6, P(3)=1 / 3, P(4)=1 / 3,\) \(P(5)=1 / 6, P(6)=0\) (b) Choose a college student at random and record gender and enrollment status: \(P(\text { female full-time })=\) \(0.56, P(\text { male full -time })=0.44, P(\text { female part-time })=\) \(0.24, P(\text { male part-time })=0.17\) (c) Deal a card from a shuffled deck: \(P(\text { clubs })=\) \(12 / 52, P\) (diamonds \()=12 / 52, P(\text { hearts })=12 / 52\) \(P(\text { spades })=16 / 52\) .

Tossing coins Imagine tossing a fair coin 3 times. (a) What is the sample space for this chance process? (b) What is the assignment of probabilities to outcomes in this sample space?

Scrabble In the game of Scrabble, each player begins by drawing 7 tiles from a bag containing 100 tiles. There are 42 vowels, 56 consonants, and 2 blank tiles in the bag. Cait chooses her 7 tiles and is surprised to discover that all of them are vowels. We can use a simulation to see if this result is likely to happen by chance. (a) State the question of interest using the language of probability. (b) How would you use random digits to imitate one repetition of the process? What variable would you measure? (c) Use the line of random digits below to perform one repetition. Copy these digits onto your paper. Mark directly on or above them to show how you determined the outcomes of the chance process. 00694 05977 19664 65441 20903 62371 22725 53340 (d) In 1000 repetitions of the simulation, there were 2 times when all 7 tiles were vowels. What conclusion would you draw?
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