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6. I have \( n \) photos in a folder on my laptop. My screensaver displays these photos and always chooses the next photo to be shown from the \( n-1 \) possible photos (excluding the one currently showing). Define the random variable \( X \) as "the number of unique pictures shown before any photo appears twice."

(a) What is the probability \( P(X=k) \) that there are exactly \( k \) different pictures before the same photo appears again (\( k=2, \ldots, n \))?

(b) I noticed that there are surprisingly few draws before I see a photo twice. The expected value or mean of a random variable is \(\sum_{k} kP(X=k)\). Numerically compute the expected value in the cases \( n=10, 50, 100 \) (you may use a computer for this if you wish).

Answer :

The random variable X represents the number of unique pictures shown before any photo appears twice in the screensaver. The probability P(X=k) can be calculated using combination and probability formulas. The expected value of X can be computed by summing all the products of k and P(X=k) for each possible value of k.

The random variable X represents the number of unique pictures shown before any photo appears twice in the screensaver. To find the probability P(X=k), we need to consider the number of ways we can choose k different pictures out of the n available pictures. The expected value of X can be computed by summing all the products of k and P(X=k) for each possible value of k.

  1. Probability P(X=k) = C(n, k) * (n-1)^(k-1) / n^k
  2. Expected value = ∑(k * P(X=k)) for k = 2 to n

Learn more about Probability here:

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