Problem 52 - Algebra and Trigonometry

Question

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 \(\frac{1}{373,248,000}\).

Step 1: Identify Total Arrangements

First, we need to determine how many different ways the monkey can arrange the six letters. Since there are six unique letters, they can be arranged in \[6! = 720\] different ways.

Step 2: Determine the Correct Arrangement

Since the only acceptable arrangement is 'MONKEY', there is exactly one correct way to spell the word among the 720 possible arrangements.

Step 3: Calculate Probability of One Correct Arrangement

The probability that the monkey spells the word correctly in one attempt is the ratio of the one correct arrangement to the total number of possible arrangements, \[\frac{1}{720}.\]

Step 4: Calculate Consecutive Correct Spellings

To find the probability of the monkey spelling 'MONKEY' correctly three consecutive times, we calculate \[\left( \frac{1}{720} \right)^3 = \frac{1}{720^3}.\]

Step 5: Compute Probability Value

Finally, compute \[720^3 = 373,248,000.\] Thus, the probability of the monkey spelling the word correctly three consecutive times is \[\frac{1}{373,248,000}.\]

Key Concepts

Factorial

The concept of a factorial is fundamental in understanding permutations and combinations in probability. A factorial, denoted by the symbol '!', refers to the product of all positive integers up to a certain number. For instance, the factorial of 6, written as \( 6! \), is calculated as:

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

This calculation is crucial when determining how many ways we can arrange a set of objects—in this case, letters. Understanding factorials is essential for problems like our monkey spelling exercise because it helps us establish the total number of possible arrangements.

Permutation

Permutations refer to the different possible arrangements of a set of objects. With a set of unique items, each different sequence of items is a permutation. In our example with the word MONKEY, each arrangement of its 6 unique letters is a different permutation. We determine the number of permutations by calculating the factorial of the number of items:

• For MONKEY, we find there are \( 6! \) or 720 permutations.

Permutations help us understand the structure of complex probability problems and are often used when the order of items is significant.

Probability of Consecutive Events

When we talk about the probability of consecutive events, we're discussing the chance that a particular event occurs back-to-back without interruption. This usually involves multiplying the probabilities of individual events happening in sequence.In our exercise, we looked for the probability that the monkey spells 'MONKEY' correctly three times in a row.

• The probability for one correct spelling is \( \frac{1}{720} \).

To find the probability of this occurring three consecutive times, we calculate:

• \( \left( \frac{1}{720} \right)^3 = \frac{1}{373,248,000} \).

Consecutive events in probability multiply the difficulty, as each subsequent event relies on the success of the prior event.

Arrangement of Letters

Arranging letters involves understanding permutations, especially when dealing with distinct letters as in the word MONKEY. Finding arrangements boils down to considering each unique order as possible.In the context of our exercise, it’s about arranging 6 letters in every possible sequence to get MONKEY. With each letter unique, each sequence truly counts—this creates a set with:

• \( 6! = 720 \) distinct arrangements.

The only correct arrangement among these is the specific permutation 'MONKEY'. Problems involving the arrangement of letters emphasize careful accounting of each unique order against what is desired—be it a single correct outcome as in spelling a particular word.