A monkey types on a 26-letter keyboard that has lowercase...

A monkey types on a 26-letter keyboard that has lowercase letters only. Each letter is chosen independently and uniformly at random from the alphabet. If the monkey types $1,000,000$ letters, what is the expected number of times the sequence "proof" appears?

Key Concepts

Independent Selections

Independent selections refer to scenarios where the outcome of one event does not affect the outcomes of other events. In the context of typing letters, each letter being chosen uniformly at random from a set means that each selection is independent, which simplifies the calculation of the overall probability of a sequence appearing.

Indicator Random Variables

Indicator random variables are simple tools used in probability to denote whether a specific event occurs (typically taking a value of 1 if the event occurs and 0 if it does not). They are particularly useful for counting occurrences of events, as their expected value directly equals the probability of that event happening.

Probability of a Specific Sequence

Calculating the probability of a specific sequence involves determining the likelihood that a given sequence of outcomes occurs in order. This is achieved by multiplying the individual probabilities of each character or event in the sequence, assuming independence among the events.

Expected Value

The expected value is a fundamental concept in probability theory that represents the average outcome of a random process when repeated many times. It is calculated by summing the products of possible outcomes and their associated probabilities, giving a measure of the central tendency for a random variable.

Linearity of Expectation

Linearity of expectation is a powerful principle stating that the expected value of a sum of random variables is equal to the sum of their expected values, regardless of whether or not the individual variables are independent. This principle simplifies the calculation of total expected occurrences by allowing the consideration of each event separately before summing the results.

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