Question

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.

Answer 1

To find the probability of selecting four vowels out of seven tiles from a bag containing 44 vowels and 56 consonants, we can use the combination formula. The probability can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Step-by-step explanation:

To determine the probability that four of the seven tiles selected from the bag are vowels, we need to find the probability of selecting four vowels out of seven.

First, we need to determine the total number of ways to select four vowels from the 44 available. This can be calculated using the combination formula: C(44, 4) = 44! / (4!(44-4)!).

Next, we need to determine the total number of ways to select three consonants from the 56 available. This can be calculated using the combination formula: C(56, 3) = 56! / (3!(56-3)!).

The total number of possible outcomes is the total number of ways to select seven tiles from 100: C(100, 7) = 100! / (7!(100-7)!).

Finally, we can calculate the probability by dividing the number of favorable outcomes (four vowels) by the total number of possible outcomes: P(Four vowels) = (C(44, 4) * C(56, 3)) / C(100, 7).

Answer 2

a. I, would think, that the question of interest is we're trying to find the probability that all of a person's tiles, after drawing seven, are vowels.
b. This is assumed to be choosing tiles without replacement (at least, that's what I'm assuming based on the introduction). What I'd do is use the numbers 00 – 99 (100 numbers total). 00 – 41 will be the numbers that mean I obtained a vowel! All others, not a vowel. Now I'll use numbers 00 – 98; if I obtained a vowel on the first one, then I'll use numbers 00 – 40; otherwise, 00 – 41 is still 'I obtained a vowel'. Now I draw a second time. Now for the third draw, the numbers will be 00 – 97. If I obtained another vowel on my second draw, 00 – 39 will be 'I obtained a vowel'; if I obtained my VERY first vowel on the second draw, 00 – 40 will be 'I obtained my vowel' for the third draw. Repeat this process seven times with the given stipulations.
c. Yow, that was a mouthful. So, using my stipulations, my numbers are:
00 (vowel); 69 (not); 40 (vowel); 59 (not); 77 (not); 19 (vowel); 66 (not)
I got three vowels in that one repetition.
d. So my statistic is p-hat, or the sample proportion = 0.002. I could conclude that THAT is the probability that someone gets all vowels after drawing seven tiles [in statistical terms, that's me using p-hat as an unbiased estimator of the parameter p].
e. I would expect to play, on average, 10000 games for that to happen. (That's the first number that came to mind; I really don't know if there's a right way to guess.)