To solve these problems, we need to calculate probabilities based on the binomial distribution with $n = 20$ and $p = 0.65$. The binomial probability mass function (PMF) is defined as:

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

The cumulative probability $P(X \leq k)$ is the sum of the PMF values up to $k$. I'll compute these for each question.

1. $P(X = 13)$

This is the probability that $X = 13$.

$P(X = 13) = \binom{20}{13} (0.65)^{13} (1 - 0.65)^{7}$

2. $P(X < 15)$

This is the cumulative probability up to $X = 14$, i.e., $P(X \leq 14)$.

$P(X < 15) = \sum_{k=0}^{14} P(X = k)$

3. $P(7 \leq X \leq 14)$

This is the sum of probabilities from $X = 7$ to $X = 14$:

$P(7 \leq X \leq 14) = \sum_{k=7}^{14} P(X = k)$

4. $P_{X < 15}(X = 13)$

This is the conditional probability:

$P_{X < 15}(X = 13) = \frac{P(X = 13)}{P(X < 15)}$

I'll compute these probabilities for you step-by-step. math import comb from scipy.stats import binom

Parameters

n = 20 p = 0.65

1. P(X = 13)

P_X_13 = comb(n, 13) * (p ** 13) * ((1 - p) ** (n - 13))

2. P(X < 15) = P(X <= 14)

P_X_less_15 = binom.cdf(14, n, p)

3. P(7 <= X <= 14)

P_7_to_14 = sum(binom.pmf(k, n, p) for k in range(7, 15))

4. P_{X < 15}(X = 13)

P_conditional = P_X_13 / P_X_less_15

Prepare results

results = { "P(X=13)": P_X_13, "P(X<15)": P_X_less_15, "P(7<=X<=14)": P_7_to_14, "P_{X<15}(X=13)": P_conditional, }

resultsHere are the computed probabilities based on the binomial distribution:

1. $P(X = 13) = 0.1844$
2. $P(X < 15) = 0.7546$
3. $P(7 \leq X \leq 14) = 0.7531$
4. $P_{X < 15}(X = 13) = 0.2444$

If you have further questions or need detailed steps for the calculations, let me know!

Here are 5 related questions for practice:

1. How would the probabilities change if $p$ were decreased to 0.5?
2. What is the probability $P(X > 15)$ for the given parameters?
3. How do you interpret the conditional probability $P_{X < 15}(X = 13)$?
4. Can you calculate the expected value and variance of $X$?
5. How would you calculate $P(X \leq 10)$ using a calculator?

Tip: For binomial probabilities, using statistical software or a calculator simplifies cumulative probability calculations significantly.