A child types the letters Q, W, E, R, T, Y, randomly producing 1000 letters in all. What is the

step 1 We are considering here a string of 1 ,000 characters, and each character has an equal chance of

A child types the letters Q, W, E, R, T, Y, randomly...

A child types the letters $\mathrm{Q}, \mathrm{W}, \mathrm{E}, \mathrm{R}, \mathrm{T}, \mathrm{Y},$ randomly producing 1000 letters in all. What is the expected number of times that the sequence QQQQ appears, counting overlaps?

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Key Concepts

Linearity of Expectation

Linearity of expectation is a principle stating that the expected value of the sum of random variables is equal to the sum of their individual expected values, regardless of any dependence between them. This principle is particularly useful for calculating the total expected number of occurrences in a process by summing the expectations of indicator variables.

Overlapping Occurrences

Counting overlapping occurrences involves identifying and accounting for instances where a particular pattern can appear in positions that share common elements. This requires careful analysis to ensure all valid, even partially overlapping, instances of the event are included in the overall count.

Expected Value

Expected value is a fundamental concept in probability that represents the long-run average or mean outcome of a random process. It is calculated by multiplying each possible outcome by its probability and then summing all these products, offering a predictive measure over numerous trials.

Indicator Random Variable

An indicator random variable is used to denote whether a specific event occurs, taking the value 1 if the event happens and 0 otherwise. This tool simplifies counting occurrences in a sequence by focusing on individual events and summing their contributions.

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