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Chapter 4: Problem 4

Consider the experiment of tossing a coin three times. a. Develop a tree diagram for the experiment. b. List the experimental outcomes. c. What is the probability for each experimental outcome?

Short Answer

Expert verified
There are 8 outcomes, each with a probability of \( \frac{1}{8} \).

Step by step solution

01

Understanding the Experiment

The experiment involves tossing a coin three times, and we need to analyze the possible outcomes. Each coin toss has two possible results: heads (H) or tails (T).
02

Constructing the Tree Diagram

Start with the first coin flip, which has two branches: H and T. For each result of the first flip, add two branches for the results of the second flip (H and T). Repeat the process for the third toss. This will result in a tree diagram that shows all the possible outcomes from all three tosses.
03

Listing the Experimental Outcomes

Look at the ends of each branch of the tree diagram. The possible outcomes are: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. There are 8 possible outcomes.
04

Calculating the Probability for Each Outcome

Since each coin toss is independent, the probability of any particular sequence of heads and tails is \( \left( \frac{1}{2} \right)^3 = \frac{1}{8} \). This is because there are three independent tosses, and the probability of heads or tails in each toss is \( \frac{1}{2} \). Therefore, the probability of each specific outcome (like HHH, HTT, etc.) is \( \frac{1}{8} \).

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Key Concepts

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

Tree Diagram
A tree diagram is a visual representation used to map out all possible outcomes of a particular experiment in probability theory. It branches out systematically to showcase the possible results at each stage. In the case of tossing a coin three times, we start with a single node representing the first coin toss. From this node, two branches appear: one for heads (H) and one for tails (T).

Each of these branches leads to another set of nodes representing the next toss, again creating two further branches for each outcome. Thus, we continue this branching process for the third and final toss. The tree diagram culminates in eight end nodes, each representing a unique sequence of outcomes, such as HHH or TTT.

Using a tree diagram helps to visually break down the complexity of the problem, making it easier to see all possible experimental outcomes. It is incredibly useful for ensuring that no potential outcome is missed when calculating probabilities.
Experimental Outcomes
Experiment outcomes refer to the final results of an experiment, laid out as all the possible scenarios that might occur. For our coin-tossing experiment, these outcomes are derived from the endpoint sequences on the tree diagram. Each sequence represents an independent series of coin results over three tosses.

The outcomes are as follows:
  • HHH
  • HHT
  • HTH
  • HTT
  • THH
  • THT
  • TTH
  • TTT
These eight unique outcomes encompass all possible ways the coins can land during the three tosses. Recognizing these outcomes is crucial, as it forms the basis of quantifying probabilities in the experiment.
Independent Events
In probability theory, events are independent if the outcome of one event does not affect the outcome of another. This principle is clear in our coin-tossing experiment: the result of each flip is independent of the previous ones.

Because of this independence, the probability of any specific sequence occurring can be determined by multiplying the probabilities of each individual event taking place. As our exercise demonstrates, the probability of getting heads (H) or tails (T) in any single toss is exactly \( \frac{1}{2} \). Thus, the probability of a specific sequence like HHT is calculated as: \[\left( \frac{1}{2} \right) \times \left( \frac{1}{2} \right) \times \left( \frac{1}{2} \right) = \frac{1}{8}.\]

Understanding independent events allows us to accurately compute the likelihood of sequences without accounting for dependencies, simplifying tasks in probability calculations.

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Most popular questions from this chapter

The Powerball lottery is played twice each week in 28 states, the Virgin Islands, and the District of Columbia. To play Powerball a participant must purchase a ticket and then select five numbers from the digits 1 through 55 and a Powerball number from the digits 1 through 42\. To determine the winning numbers for each game, lottery officials draw 5 white balls out of a drum with 55 white balls, and 1 red ball out of a drum with 42 red balls. To win the jackpot, a participant's numbers must match the numbers on the 5 white balls in any order and the number on the red Powerball. Eight coworkers at the ConAgra Foods plant in Lincoln, Nebraska, claimed the record \(\$ 365\) million jackpot on February \(18,2006,\) by matching the numbers \(15-17-43-44-49\) and the Powerball number \(29 .\) A variety of other cash prizes are awarded each time the game is played. For instance, a prize of \(\$ 200,000\) is paid if the participant's five numbers match the numbers on the 5 white balls (Powerball website, March 19,2006 ). a. Compute the number of ways the first five numbers can be selected. b. What is the probability of winning a prize of \(\$ 200,000\) by matching the numbers on the 5 white balls? c. What is the probability of winning the Powerball jackpot?

Consider the experiment of rolling a pair of dice. Suppose that we are interested in the sum of the face values showing on the dice. a. How many sample points are possible? (Hint: Use the counting rule for multiple-step experiments.) b. List the sample points. c. What is the probability of obtaining a value of \(7 ?\) d. What is the probability of obtaining a value of 9 or greater? e. Because each roll has six possible even values \((2,4,6,8,10, \text { and } 12)\) and only five possible odd values \((3,5,7,9, \text { and } 11),\) the dice should show even values more often than odd values. Do you agree with this statement? Explain. f. What method did you use to assign the probabilities requested?

Consider the experiment of selecting a playing card from a deck of 52 playing cards. Each card corresponds to a sample point with a \(1 / 52\) probability. a. List the sample points in the event an ace is selected. b. List the sample points in the event a club is selected. c. List the sample points in the event a face card (jack, queen, or king) is selected. d. Find the probabilities associated with each of the events in parts (a), (b), and (c).

An experiment has four equally likely outcomes: \(E_{1}, E_{2}, E_{3},\) and \(E_{4}\). a. What is the probability that \(E_{2}\) occurs? b. What is the probability that any two of the outcomes occur (e.g., \(E_{1}\) or \(E_{3}\) )? c. What is the probability that any three of the outcomes occur (e.g., \(E_{1}\) or \(E_{2}\) or \(E_{4}\) )?

A local bank reviewed its credit card policy with the intention of recalling some of its credit cards. In the past approximately \(5 \%\) of cardholders defaulted, leaving the bank unable to collect the outstanding balance. Hence, management established a prior probability of .05 that any particular cardholder will default. The bank also found that the probability of missing a monthly payment is .20 for customers who do not default. Of course, the probability of missing a monthly payment for those who default is 1 a. Given that a customer missed one or more monthly payments, compute the posterior probability that the customer will default. b. The bank would like to recall its card if the probability that a customer will default is greater than \(.20 .\) Should the bank recall its card if the customer misses a monthly payment? Why or why not?
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