To find the probability of exactly $k = 3$ successes in a binomial distribution, we use the binomial probability formula:

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

where:

• $n = 55$ (sample size)
• $k = 3$ (desired number of successes)
• $p = 0.36$ (probability of success)
• $1 - p = 0.64$ (probability of failure)
• $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ (binomial coefficient)

Step 1: Compute the Binomial Coefficient

$\binom{55}{3} = \frac{55!}{3!(55-3)!} = \frac{55!}{3!52!}$

Step 2: Compute the Probability

$P(X = 3) = \binom{55}{3} (0.36)^3 (0.64)^{52}$

Let's calculate this value. math

Given values

n = 55 # sample size k = 3 # desired successes p = 0.36 # probability of success q = 1 - p # probability of failure

Compute binomial coefficient

binom_coeff = math.comb(n, k)

Compute probability

probability = binom_coeff * (p ** k) * (q ** (n - k))

Round to four decimal places

probability_rounded = round(probability, 4) probability_roundedThe computed probability is approximately 0.0000 when rounded to four decimal places. This suggests that the probability of getting exactly three successes out of 55 trials with a success probability of 0.36 is extremely small.

Would you like a breakdown of the calculation, or do you have any questions?

Here are five related questions you might find interesting:

1. How does the probability change if the success probability increases to 0.5?
2. What is the expected number of successes in this distribution?
3. How do we compute the probability of at most three successes?
4. How does the binomial distribution approximate a normal distribution in large samples?
5. What is the variance of this binomial distribution?

Tip: When dealing with very low probabilities, consider using a Poisson approximation to simplify calculations!