Some Questions Based on Discrete Probability Distributions

Some Questions Based on Discrete Probability Distributions

Data Science and A.I. Lecture Series

 

Problem 1

2 bad articles are mixed with 5 good ones. Find the probability distribution of the number of bad articles if 2 articles are drawn at random.

Let \( X \) be the number of bad articles drawn. Possible values: \( X = 0, 1, 2 \).

\[
P(X = 0) = \frac{\binom{5}{2}}{\binom{7}{2}} = \frac{10}{21}
\]

\[
P(X = 1) = \frac{\binom{2}{1} \binom{5}{1}}{\binom{7}{2}} = \frac{10}{21}
\]

\[
P(X = 2) = \frac{\binom{2}{2}}{\binom{7}{2}} = \frac{1}{21}
\]

Problem 2

Given the probability distribution:

X P(X)
0 \( \frac{1}{10} \)
1 \( \frac{3}{10} \)
2 \( \frac{1}{2} \)
3 \( \frac{1}{10} \)

Let \( Y = X^2 + 2X \). Find the probability distribution of \( Y \).

Computed values of \( Y \):

  • If \( X = 0 \), then \( Y = 0 \).
  • If \( X = 1 \), then \( Y = 3 \).
  • If \( X = 2 \), then \( Y = 8 \).
  • If \( X = 3 \), then \( Y = 15 \).

Problem 3

An urn contains 3 white and 4 red balls. 3 balls are drawn one by one with replacement. Find the probability distribution of the number of red balls.

Let \( X \) be the number of red balls drawn. Possible values: \( X = 0, 1, 2, 3 \).

\[
P(X = 0) = \left(\frac{3}{7}\right)^3 = \frac{27}{343}
\]

\[
P(X = 1) = 3 \times \left(\frac{4}{7} \times \frac{3}{7} \times \frac{3}{7}\right) = \frac{108}{343}
\]

\[
P(X = 2) = 3 \times \left(\frac{4}{7} \times \frac{4}{7} \times \frac{3}{7}\right) = \frac{144}{343}
\]

\[
P(X = 3) = \left(\frac{4}{7}\right)^3 = \frac{64}{343}
\]

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Some Questions on Discrete Probability Distributions

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