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. Can we use a binomial distribution to approximate this probability? Justify your answer.
The Correct Answer and Explanation is:
Correct Answer:
No, we cannot use a binomial distribution to accurately approximate this probability because the selection of Scrabble tiles is done without replacement, which violates a key condition of the binomial model.
Explanation
In probability, the binomial distribution is used to model the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (commonly labeled success and failure), and the probability of success remains constant throughout.
To determine whether the binomial distribution can be used in Cait’s situation, we must examine the key assumptions of the binomial model:
- Fixed number of trials: Cait draws 7 tiles. This condition is met.
- Only two outcomes per trial: Each tile can be a vowel (success) or not a vowel (failure). This condition is also satisfied.
- Probability of success stays the same: This is where the problem arises. Since Cait draws tiles without replacement, the total number of tiles in the bag changes with each draw. For example, if the first tile is a vowel, there are now fewer vowels and fewer total tiles in the bag. This changes the probability of drawing another vowel on the next trial.
- Trials are independent: In this case, each draw affects the next one. Since tiles are not replaced, the outcome of one draw influences the probabilities of subsequent draws. This means the trials are not independent.
Because the third and fourth conditions are violated, we cannot use a binomial distribution. Instead, a more appropriate method would be the hypergeometric distribution, which is specifically designed to model the probability of successes in draws without replacement from a finite population.
Therefore, the probability that Cait draws 7 vowels from the bag should be calculated using a hypergeometric model, not a binomial one.
