Problem 62 - Algebra and Trigonometry

Question

Making Words \(\quad\) A researcher claims that she has taught a monkey to spell the word MONKEY using the five wooden letters \(E, O, K, M, N, Y\) If the monkey has not actually learned anything and is merely arranging the blocks randomly, what is the probability that he will spell the word correctly three consecutive times?

Step-by-Step Solution

Answer

The probability is \( \frac{1}{373248000} \).

Step 1: Determine Total Arrangements

First, determine the total number of ways the monkey can arrange the six letters. Since all the letters are unique, the number of arrangements is given by the factorial of the number of letters. Thus, the total arrangements is \( 6! = 720 \).

Step 2: Determine Successful Outcomes

The word 'MONKEY' is one specific arrangement of these letters. There is only 1 successful way to arrange them in the correct order to spell 'MONKEY'.

Step 3: Calculate Probability of One Correct Arrangement

The probability of spelling the word 'MONKEY' correctly in one random arrangement is the ratio of successful outcomes to total outcomes: \( \frac{1}{720} \).

Step 4: Determine Probability of Three Consecutive Successes

Since the arrangements are independent, the probability that the word is spelled correctly three times in a row is the probability of one success raised to the third power: \( \left(\frac{1}{720}\right)^3 = \frac{1}{373248000} \).

Key Concepts

Factorial

In mathematics, a factorial is a simple yet powerful concept, especially when dealing with permutations and combinations. The factorial of a non-negative integer number is the product of all positive integers less than or equal to that number. For example, if you want to find the factorial of 6, denoted as \( 6! \), you will multiply together 6, 5, 4, 3, 2, and 1. The calculation looks like this:

• \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \)

The factorial function grows extremely fast with larger numbers, which is why calculators and computers are often used to handle them. In the case of arranging letters or objects, each distinct sequence is a permutation and is calculated using a factorial. The concept of factorial helps determine how many ways you can arrange a set of items, which is a foundational principle in probability.

Permutations

Permutations are all the different arrangements or sequences that you can create using a certain set of items. When dealing with permutations, the order of the items is crucial. This means that swapping two items provides a different permutation.

To better understand permutations, consider the example of the word "MONKEY". If we have the six unique letters E, O, K, M, N, Y, the number of permutations of these letters is computed using the factorial of 6:

• \( 6! = 720 \)

Among these 720 different arrangements, only one spells the word "MONKEY" correctly.

Permutations play a vital role in calculating probabilities when you need to determine how likely it is for random arrangements to result in a specific sequence. The concept aids in breaking down complex problems by understanding all potential sequences.

Independent Events

The concept of independent events in probability refers to scenarios where the outcome of one event does not influence the outcome of another. This is a fundamental concept when calculating probabilities, as it allows us to combine probabilities of separate events to find the likelihood of all occurring together.

Consider the exercise where the monkey attempts to spell "MONKEY" correctly three times in a row. Each attempt is independent of the others, meaning spelling it correctly once doesn’t affect the next try.

• The probability of the monkey spelling "MONKEY" correctly in one try is \( \frac{1}{720} \).
• To find the probability of this happening three consecutive times, you multiply the probability of a single success by itself twice more, which gives you:
• \( \left(\frac{1}{720}\right)^3 = \frac{1}{373248000} \)

Understanding independent events helps in determining outcomes across multiple trials or activities, especially in complex probability calculations.