Let's break down the problem step by step:

Given:

• $X$ has possible values $\{1, 2, 3, 4, 5\}$
• The probability mass function $p_X(x)$ is as follows:

$x$	1	2	3	4	5
$p_X(x)$	$\frac{1}{7}$	$\frac{1}{14}$	$\frac{3}{14}$	$\frac{2}{7}$	$\frac{2}{7}$

(a) Calculate $P(X = 3)$:

This is the probability mass function evaluated at $X = 3$: $P(X = 3) = p_X(3) = \frac{3}{14}$

(b) Calculate $P(X < 3)$:

This is the probability that $X$ takes on a value less than 3. The possible values less than 3 are 1 and 2, so we sum the probabilities for $X = 1$ and $X = 2$: $P(X < 3) = P(X = 1) + P(X = 2) = \frac{1}{7} + \frac{1}{14}$ $P(X < 3) = \frac{2}{14} + \frac{1}{14} = \frac{3}{14}$

(c) Calculate $P(X < 4 \mid X > \frac{16}{38})$:

First, simplify $\frac{16}{38}$, which is equivalent to $\frac{8}{19}$. Since $X$ can only take the values 1, 2, 3, 4, or 5, and the smallest probability for $X$ is $\frac{1}{14} \approx 0.071$, all values of $X$ exceed $\frac{8}{19} \approx 0.421$.

Therefore, this condition $X > \frac{16}{38}$ includes all possible values for $X$. So this conditional probability becomes: $P(X < 4 \mid X > \frac{16}{38}) = P(X < 4)$ Now, calculate $P(X < 4)$, which is the sum of the probabilities for $X = 1, 2, 3$: $P(X < 4) = P(X = 1) + P(X = 2) + P(X = 3) = \frac{1}{7} + \frac{1}{14} + \frac{3}{14}$ $P(X < 4) = \frac{2}{14} + \frac{1}{14} + \frac{3}{14} = \frac{6}{14} = \frac{3}{7}$

Thus, the conditional probability is: $P(X < 4 \mid X > \frac{16}{38}) = \frac{3}{7}$

Final Answers:

• (a) $P(X = 3) = \frac{3}{14}$
• (b) $P(X < 3) = \frac{3}{14}$
• (c) $P(X < 4 \mid X > \frac{16}{38}) = \frac{3}{7}$

Would you like more details or have any questions? Here are 5 related questions to extend the discussion:

1. How do we compute conditional probabilities in general?
2. What is the total probability for all possible values of $X$?
3. Can you calculate the expectation $E(X)$ using this probability mass function?
4. How would the calculations change if one of the probabilities were adjusted?
5. What is the significance of $X > \frac{16}{38}$ in this case?

Tip: Always verify that the sum of probabilities from a probability mass function (PMF) equals 1 to ensure it's valid.