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Chapter 5: Problem 61

Coin (Example 14) Imagine flipping three fair coins. a. What is the theoretical probability that all three come up heads? b. What is the theoretical probability that the first toss is tails AND the next two are heads?

Short Answer

Expert verified
a. The theoretical probability that all three come up heads is 0.125 or 1/8. b. The theoretical probability that the first toss is tails AND the next two are heads is also 0.125 or 1/8.

Step by step solution

01

Calculate Theoretical Probability of All Heads

The outcome of each coin toss is independent of the others. Therefore, the probability of getting three heads is just the product of the probabilities of getting a head on each individual toss. This is \(0.5 * 0.5 * 0.5 = 0.125\) or \(1/8\).
02

Calculate Theoretical Probability of First Toss Tails and Next Two Heads

Again, the outcome of each coin toss is independent of the others. Therefore, the probability of getting a tail on the first toss and a head on the second and third tosses is the product of the probabilities of these individual outcomes. This is \(0.5 * 0.5 * 0.5 = 0.125\) or \(1/8\).

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Key Concepts

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

Independent Events
In probability theory, understanding independent events is crucial. Two or more events are said to be independent if the outcome of one event does not influence or affect the outcome of another.

When flipping a fair coin, each flip represents an independent event. This is because the outcome of one flip (like getting heads or tails) does not affect the outcome of any subsequent flips.
  • If you flip a coin once and get heads, the probability of getting heads on the second flip remains \(0.5\).
  • Each coin toss is treated as a separate event that doesn't rely on previous results.
  • Independence means that the entire process resets with every new flip.
This concept is essential when calculating probabilities involving multiple coin tosses.
Coin Toss Probability
The probability of an individual coin toss in a fair coin is straightforward. Since there are two possible outcomes on each toss鈥攈eads or tails鈥攖he probability for each outcome is \(0.5\).
  • If you're interested in finding the probability of multiple specific outcomes, like three heads in a row, you multiply the probabilities for each individual event.
  • For example, to find the probability of getting heads three times in a row: \( 0.5 \times 0.5 \times 0.5 = 0.125\). This gives a probability of \(1/8\).
Similarly, if you want to compute the probability of the first toss being tails followed by two heads, the procedure remains the same:
  • First toss is tails: \(0.5\)
  • Second toss is heads: \(0.5\)
  • Third toss is heads: \(0.5\)
  • Thus, the total probability: \( 0.5 \times 0.5 \times 0.5 = 0.125\) or \(1/8\).
Both situations reflect how individual probabilities multiply to form the total probability in sequences of independent events.
Step-by-Step Solution
The step-by-step solution provided follows a logical and transparent method to solve the problem. Each coin toss is recognized as an independent event with probabilities that combine through multiplication.

For part (a), the aim is to find the probability of all three coins showing heads:
  • Step 1 involves the recognition of independence, allowing us to multiply probabilities.
  • Multiply the individual probabilities: \\(0.5\), \(0.5\), and \(0.5\) to get \(0.125\) or \(1/8\).
For part (b), the problem changes slightly by specifying a sequence: first tails, followed by two heads.
  • Again, you start by acknowledging the independence of events.
  • The probability of the first coin landing tails is \(0.5\). The subsequent flips being heads also have probabilities of \(0.5\) each.
  • Therefore, multiply these probabilities: \(0.5 \times 0.5 \times 0.5 = 0.125\) or \(1/8\).
This step-by-step breakdown simplifies the calculation process and reaffirms the principle of independent events in computing probabilities.

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Most popular questions from this chapter

Independent Variables Use your general knowledge to label the following pairs of variables as independent or associated. Explain. a. For a sample of adults, gender and ring size. b. The outcome on rolls of two separate, fair dices.

Random Assignment of Professors A study randomly assigned students attending the Air Force Academy to different professors for Calculus I, with equal numbers of students assigned to each professor. Some professors were experienced, and some were relatively inexperienced. Suppose the names of the professors are Peters, Parker, Diaz, Nguyen, and Black. Suppose Diaz and Black are inexperienced and the others are experienced. The researchers reported that the students who had the experienced teachers for Calculus I did better in Calculus II. (Source: Scott E. Carrell and James E. West, Does Professor Quality Matter? Evidence from Random Assignment of Students to Professors, 2010 ) a. List the equally likely outcomes that could occur for assignment of one student to a professor. b. Suppose the event of interest, event \(\mathrm{A}\), is that a teacher is experienced. List the outcomes that make up event \(\mathrm{A}\). c. What is the probability that a student will be assigned to an experienced teacher? d. List the outcomes in the complement of event \(\mathrm{A}\). Describe this complement in words. e. What is the probability that a student will be assigned to an inexperienced teacher?

Online Test An online test consists of 20 multiple-choice questions. Each of the 20 answers is either right or wrong. Suppose the probability that an examinee gets fewer than 5 answers correct is \(0.42\) and the probability that an examinee gets from 5 to 12 (inclusive) answers correct is \(0.38\). Find the probability that an examinee gets: a. More than 12 answers correct b. At most 12 answers correct c. 50 or more answers correct d. Which of the two events in \(\mathrm{a}-\mathrm{c}\) are complements of each other, and why?

Simulation a. Explain how you could use digits from a random number table to simulate rolling a fair eight-sided die with outcomes \(1,2,3,4,5,6,7\), and 8 equally likely. Assume that you want to know the probability of getting a 1 . b. Carry out your simulation, beginning with line 5 of the random number table in Appendix A. Perform 20 repetitions of your trial. Using your results, report the empirical probability of getting a 1, and compare it with the theoretical probability of getting a 1 .

Playing Cards Refer to Exercise \(5.11\) for information about cards. If you draw 1 card randomly from a standard 52 -card playing deck, what is the probability that it will be: a. A black card? b. A diamond? c. A face card (jack, queen, or king)? d. A nine? e. A king or queen?
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