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
The Correct Answer and Explanation is:
(a) State the question of interest using the language of probability:
What is the probability that a player randomly draws 7 tiles from a standard Scrabble bag and all 7 tiles turn out to be vowels?
(b) How would you use random digits to imitate one repetition of the process? What variable would you measure?
To simulate one repetition, we can represent the 100 Scrabble tiles using digits. Since there are 42 vowels, 56 consonants, and 2 blanks, we can assign:
- Digits 00 to 41 for vowels (42 outcomes),
- Digits 42 to 97 for consonants (56 outcomes),
- Digits 98 and 99 for blanks (2 outcomes).
We will select 7 random two-digit numbers from a random digit table. For each digit pair, we check whether it falls in the vowel range (00 to 41). The variable we measure is the number of vowels out of the 7 selected tiles.
(c) Use the line of random digits below to perform one repetition.
Digits:00694 05977 19664 65441 20903 62371 22725 53340
Group into two-digit numbers:
00, 69, 40, 59, 77, 19, 66, 46, 54, 41, 20, 90, 36, 23, 71, 22, 72, 55, 33, 40
Now pick the first 7 valid two-digit numbers:
00 (vowel), 69 (consonant), 40 (vowel), 59 (consonant), 77 (consonant), 19 (vowel), 66 (consonant)
Result: 3 vowels and 4 consonants out of 7 tiles.
(d) In 1000 repetitions of the simulation, there were 2 times when all 7 tiles were vowels. What conclusion would you draw?
If all 7 tiles are vowels only 2 times out of 1000 simulations, this gives a probability of 2 out of 1000, or 0.002. This result is extremely unlikely to occur by chance alone. Therefore, we would conclude that Cait’s draw of 7 vowels is highly unusual and probably not due to pure chance.
