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A binomial distribution is appropriate for a random variable that represents the number of successes in a fixed number of Bernoulli trials, which are independent with identical probability of success on each trial. In this case, however, x does not represent the number of successes (songs by the particular artist) in a fixed number of trials (songs played). Instead, x represents the number of trials until success occurs, and the trials do not have a fixed number. Therefore, the binomial distribution is not appropriate for this problem.

Since the binomial distribution is not appropriate for this problem, let's consider another probability distribution, the geometric distribution. A geometric distribution is appropriate for a random variable that represents the number of trials until the first success (song by the particular artist) occurs. The probability mass function (PMF) of a geometric distribution is given by: \[p(x) = (1-p)^{(x-1)}p\] where p is the probability of success on each trial, which in our case is the probability of selecting a song by the particular artist (\(\frac{8}{100}\)). Now, let's find the required probabilities using the geometric distribution.

\(p(4)\) represents the probability that the first song by the particular artist is played after exactly 4 songs. Using the PMF of a geometric distribution, we can calculate \(p(4)\) as follows: \[p(4)= (1-\frac{8}{100})^{(4-1)}\frac{8}{100} = (0.92)^3(0.08) \approx 0.0575\]

\(P(x \leq 4)\) represents the probability that the first song by the particular artist is played within the first 4 songs. To calculate this probability, we can sum up the individual probabilities for the events x=1, x=2, x=3, and x=4: \[P(x\leq 4) = p(1) + p(2) + p(3) + p(4)\] Using the PMF of a geometric distribution: \[P(x \leq 4) = (1-p)^{0}p + (1-p)^{1}p + (1-p)^{2}p + (1-p)^{3}p\] Substituting the value of p (\(\frac{8}{100}\)): \[P(x \leq 4) = (1-0.08)^0 (0.08) + (0.92)^1 (0.08) + (0.92)^2 (0.08) + (0.92)^3 (0.08) \approx 0.2819\]

\(P(x>4)\) represents the probability that the first song by the particular artist is played after the first 4 songs. To calculate this probability, we can use the complement rule: \[P(x>4) = 1 - P(x\leq 4)\] Using the previously calculated value for \(P(x \leq 4)\): \[P(x>4) = 1 - 0.2819 \approx 0.7181\]

\(P(x \geq 4)\) represents the probability that the first song by the particular artist is played on or after the 4th song. We can calculate this probability by summing up the individual probabilities for the events \(x=4, x=5, x=6, \dots\). However, we can use the following property to simplify the calculation: \[P(x \geq 4) = P(x\geq 3) - P(x=3)\] Using the PMF of a geometric distribution: \(P(x\geq 3) = 1 - [ p(1) + p(2) ] = 1 - [ (1-p)^0 p + (1-p)^1 p ]\) Substituting the value of p (\(\frac{8}{100}\)): \(P(x\geq 3) = 1 - [(1-0.08)^0 (0.08) + (0.92)^1 (0.08)] = 1 - [0.08 + 0.0752] = 0.8448\) Now, we can calculate \(p(3)\) using the PMF of a geometric distribution: \[p(3) = (1-p)^{(3-1)}p = (0.92)^2(0.08) \approx 0.0673\] Finally, we can find \(P(x \geq 4)\): \[P(x \geq 4) = P(x\geq 3) - P(x=3) = 0.8448 - 0.0673 \approx 0.7775\]

a) \(p(4) \approx 0.0575\): This represents the probability that the first song by the particular artist is played exactly after 3 other songs, as the 4th song. b) \(P(x\leq4) \approx 0.2819\): This denotes the probability that the first song by the particular artist is played within the first 4 songs. c) \(P(x>4) \approx 0.7181\): This probability suggests that the first song by the particular artist is played after the first 4 songs. d) \(P(x \geq 4) \approx 0.7775\): This represents the probability that the first song by the particular artist is played on or after the 4th song. The difference between these probabilities is based on the events they represent. Each probability represents a different way of looking at when the first song by the particular artist will be played. For example, \(P(x\leq4)\) accounts for the number of songs played before the first song by the artist, while \(P(x\geq4)\) accounts for the number of songs played from the fourth song onwards until the first song by the artist is played.