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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?

Step by step solution

01

Stating the Probability Question

The question of interest is: What is the probability of drawing 7 vowels from a bag containing 42 vowels, 56 consonants, and 2 blank tiles, in a total of 100 tiles? We want to determine if this event is likely to occur by chance.

02

Designing the Simulation

To simulate this scenario, let's represent each tile with a digit: 0-41 represent vowels, 42-97 represent consonants, and 98-99 represent blank tiles. We simulate drawing 7 tiles by selecting 7 digits at random from the range 0 to 99. The variable we measure is the count of vowels among the 7 selected tiles.

03

Using Random Digits for One Repetition

We use the provided line of random digits: 00694 05977 19664 65441 20903 62371 22725 53340. Mark and analyze each group of 7 digits as one trial. They represent tile draws: - First draw: 0069405, vowels: 0069405 (all vowels since they fall between 0 and 41). - First repetition has 3 vowels drawn.

04

Analyzing Long-Run Simulation Results

According to the provided 1000 simulations, only 2 out of 1000 resulted in all tiles being vowels. Calculate the probability: Probability = (Number of successful trials) / (Total trials) = 2/1000 = 0.002.

05

Conclusion from Simulation Results

The probability of drawing 7 vowels by chance is 0.2%. This extremely low probability suggests that Cait's result is highly unlikely to occur by random chance alone.

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Key Concepts

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

Scrabble Tiles

In Scrabble, each player draws 7 tiles from a bag originally holding 100 tiles. This bag consists of 42 vowels, 56 consonants, and 2 blank tiles. The goal for Cait is to draw tiles that allow her to form words and earn points. In this scenario, Cait unexpectedly draws 7 vowels. This is a rare event that drives us to question its likelihood using probability. The makeup of the Scrabble tiles offers a practical model for probability exercises. Knowing the composition of these tiles aids in understanding probabilities and simulating random events. This acts as a foundation for analyzing the chance of drawing all vowels among a broader context of vowel, consonant, and blank tile distributions.

Random Digits

Random digits, in this exercise, simulate the action of drawing tiles from a Scrabble bag. Each tile is given a range of digits:

• 0-41 for vowels
• 42-97 for consonants
• 98-99 for blank tiles

These digits mimic the physical act of grabbing a tile randomly from the bag, with a perfect representation of real-world probabilities since "digits" are selected without bias. During the simulation, we use a pre-arranged sequence of digits to simulate draws. This method allows for a straightforward and replicable process to predict outcomes. The process prioritizes randomness, sticking to statistical probability principles while ensuring simplicity and accessibility to simulate each event.

Vowel Probability

Calculating the probability of pulling 7 vowels in a row from a Scrabble bag is crucial to understanding the odds of such a rare event. Given there are 42 vowels out of 100 tiles, the task involves recognizing how this ratio dramatically affects outcome likelihood. In probability, each draw affects subsequent choices. The probability of selecting a vowel continuously diminishes as tiles are drawn.

• First vowel probability: 42/100
• Second vowel probability: 41/99
• And so forth, reducing the number of vowels and total tiles

Ultimately, calculating these compounded probabilities addresses how surprisingly unlikely Cait's draw is.

Simulation Analysis

Simulation is a powerful tool within probability to predict possible outcomes by running repetitive trials. By observing the outcome in controlled scenarios, such as the 1000 repetitions of drawing Scrabble tiles, we can gain insights into rare events. In this exercise, achieving a complete set of vowels only twice in 1000 trials highlights Cait's remarkable draw. Through simulations, we derived a probability of 0.2% for drawing 7 vowels, emphasizing how seldom such occurrences are. Simulations help demystify probability for learners by visually presenting outcomes through repetitive iterations. By modeling the scenario numerous times, simulations allow students to build an understanding of probability's intricacies without assumptive calculations.