Question

Scrabble In the game of Scrabble, each player begins by randomly selecting $7$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 want to perform a simulation to determine the probability that a player will randomly select $7$vowels.

a. Describe how you would use a table of random digits to carry out this simulation.

b. Perform one trial of the simulation using the random digits given. Copy the digits onto your paper and mark directly on or above them so that someone can follow what you did.

c. In $2$of the $1000$trials of the simulation, all $7$tiles were vowels. Does this result give convincing evidence that the bag of tiles was not well mixed?

Short answer

a. Each vowel tile is assigned a unique number ranging from $00$to $41$, each consonant tile is assigned a unique number ranging from $42$to $97$, and the two blank tiles are assigned $98$and $99$Now we'll pick a row from the table of random digits.

b. We'll now choose the first two-digit number. Then choose the equivalent tile and repeat until you've chosen seven different tiles. So there you have it.

c. There are so few trials in which all seven tiles are vowels, it's likely that all seven tiles are vowels, and so there's compelling evidence that the bag wasn't thoroughly mixed.

Step-by-step solution

Part (a) Step 1 : Given Information

We have to describe how you would use a table of random digits to carry out this simulation.

Part (a) Step 2 : Simplification

The question includes a doodle game in which each player must choose seven tiles from a bag of tiles. We now have $100$tiles, which include $42$vowels, $56$consonants, and two blank tiles. $7$tiles are chosen from a total of $100$. As a result, each vowel tile is assigned a unique number ranging from $00$to $41$, each consonant tile is assigned a unique number ranging from $42$to $97$, and the two blank tiles are assigned $98$and $99.$Now we'll pick a row from the table of random digits. Then we choose the appropriate tile. We'll keep going until you've chosen seven tiles.

Part (b) Step 1 : Given Information

We have to perform one trial of the simulation using the random digits given.

Part (b) Step 2 : Simplification

The question includes a doodle game in which each player must choose seven tiles from a bag of tiles.

We now have $100$tiles, which include $42$vowels, $56$consonants, and two blank tiles. $7$tiles are chosen from a total of $100$. As a result, each vowel tile is assigned a unique number ranging from $00$to $41$, each consonant tile is assigned a unique number ranging from $42$to $97$, and the two blank tiles are assigned $98$and $99$.

We now select a row from the random digits database as follows:

$00694,05977,19664,65441,20903,62371,22725,53340$

We'll now choose the first two-digit number. Then choose the equivalent tile and repeat until you've chosen seven different tiles. So there you have it.

$00$⇒ Select tile $00$

$69$⇒ Select tile $69$

$40$⇒ Select tile40

$59$⇒ Select tiles $59$

$77$⇒ Select tile $77$

$19$⇒ Select tile $19$

$66$⇒ Select tile$66$

As a result, we can see that the sample contains tiles $00,69,40,59,77,19,6600,69,40,59,77,19,66$, but only tiles $00,40$and $19$are vowel tiles.

Part (c) Step 1 : Given Information

We have to explain does thus result gives convincing evidence that the bag of tiles was not well mixed.

Part (c) Step 2 : Simplification

The question includes a doodle game in which each player must choose seven tiles from a bag of tiles. We now have $100$tiles, which include $42$vowels, $56$consonants, and two blank tiles. Given that two out of every$100$ simulation trials result in all seven tiles being vowels. Because there are so few trials in which all seven tiles are vowels, it's likely that all seven tiles are vowels, and so there's compelling evidence that the bag wasn't
thoroughly mixed.