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Hank and Rosa play the following game: A bag has 15

tiles in it, each with a letter from the phrase probability game on it. Hank and Rosa take turns drawing a tile, recording the letter, and placing the tile back in the bag. Hank earns a point if he draws a vowel. Rosa earns a point if she draws a consonant. They decide that the letter y can be a vowel or a consonant. Which statement best explains whether or not the game is fair?

1 answer

To determine whether the game is fair, we need to analyze the composition of the letters in the phrase "probability game" and understand how many tiles represent vowels and consonants.

First, let's identify the letters and count them:

  • The phrase "probability game" consists of: p, r, o, b, a, b, i, l, i, t, y, g, a, m, e.

Now, we can categorize the letters as vowels and consonants:

  • Vowels: a, e, i, o, (and y, which can also be considered a vowel)
  • Consonants: p, r, b, b, l, t, g, m (and y, which can also be considered a consonant)

Now, we count each group:

  1. Total Letters:
    • There are 15 letters in total.
  2. Vowels (counting 'y' as a vowel):
    • The total vowels are: a (2), e (1), i (2), o (1), y (1) = 2 + 1 + 2 + 1 + 1 = 7 vowels.
  3. Consonants (counting 'y' as a consonant):
    • The consonants are: p (1), r (1), b (2), l (1), t (1), g (1), m (1), y (1) = 1 + 1 + 2 + 1 + 1 + 1 + 1 = 8 consonants.

However, if we consider 'y' as a vowel only:

  • Vowels: a (2), e (1), i (2), o (1), y (1) = 2 + 1 + 2 + 1 + 1 = 7 vowels.
  • Consonants: p (1), r (1), b (2), l (1), t (1), g (1), m (1) = 1 + 1 + 2 + 1 + 1 + 1 = 8 consonants.

So now we can summarize:

  • The total chances for Hank (with 'y' as a vowel): 7 vowels.
  • The total chances for Rosa (with 'y' as a consonant): 8 consonants.

Hank's chance of drawing a vowel = 7/15, Rosa's chance of drawing a consonant = 8/15.

Given these probabilities, Rosa has a slightly better chance of scoring a point than Hank does, specifically:

  • Hank has a probability of approximately 46.67% (7 out of 15),
  • Rosa has a probability of approximately 53.33% (8 out of 15).

Thus, we can conclude that:

The game is not fair, since Rosa (drawing consonants) has a higher probability of scoring than Hank (drawing vowels), when 'y' is considered only as a consonant. If 'y' is treated as both, it still favors the consonants slightly. Therefore, the imbalance in scoring opportunities makes it an unfair game.