Question: Following is the probability distribution of a random variable that represents the number of…

Following is the probability distribution of a random variable that represents the number of extracurricular activities a college freshman participates in.

\[ \begin{array}{c|ccccc} x & 0 & 1 & 2 & 3 & 4 \\ \hline P(x) & 0.05 & 0.13 & 0.45 & 0.21 & 0.16 \\ \end{array} \]

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Part: 0 / 3

Part 1 of 3

(a) Find the probability that a student participates in exactly three activities.

The probability that a student participates in exactly three activities is \(\boxed{\rule{1cm}{0.15mm}}\).

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Part: 1 / 3

Part 2 of 3

(b) Find the probability that a student participates in less than three activities.

The probability that a student participates in less than three activities is \(\boxed{\rule{1cm}{0.15mm}}\).

Part: 2 / 3

Part 3 of 3

(c) Find the probability that a student participates in at most three activities.

The probability that a student participates in at most three activities is \(\boxed{\rule{1cm}{0.15mm}}\).

Solution

The image shows a probability distribution for a random variable representing the number of extracurricular activities a college freshman participates in. The probabilities associated with each number of activities are given. Let’s solve each part: (a) Find the probability that a student participates in exactly three activities. The probability given for exactly three activities \( x = 3 \) is \( P(3) = 0.21 \). (b) Find the probability that a student participates in less than three activities. Add the probabilities for \( x = 0 \), \( x = 1 \), and \( x = 2 \): \[ P(x < 3) = P(0) + P(1) + P(2) = 0.05 + 0.13 + 0.45 = 0.63 \] (c) Find the probability that a student participates in at most three activities. Add the probabilities for \( x = 0 \), \( x = 1 \), \( x = 2 \), and \( x = 3 \): \[ P(x \leq 3) = P(0) + P(1) + P(2) + P(3) = 0.05 + 0.13 + 0.45 + 0.21 = 0.84 \]