Quackers decides to measure the number of quacks he makes in half an hour. It turns out that it is 18. He wants to calculate the probability that he would have made that many quacks if the underlying true probability of a quack in any minute was 55%. How should he do this?

Question

BlackTom AI · Accepted Answer

Number of quacks in 30 minutes follows a Binomial(n = 30, p = 0.55), so the probability of exactly 18 quacks is given by the binomial probability mass function and in R is computed with dbinom(x = 18, size = 30, prob = 0.55).

1. Relevant concepts
- Each minute is a Bernoulli trial (quack or no quack) with success probability p = 0.55.
- The total number of successes in 30 independent minutes is Binomial(n = 30, p = 0.55).
- The probability of exactly k successes is P(X = k) = C(n,k) p^k (1 − p)^(n − k).
- In R, dbinom(x, size, prob) returns this binomial probability mass.

2. Step-by-step reasoning
- Identify n = 30, k = 18, p = 0.55.
- Apply the binomial formula: P(X = 18) = C(30,18) (0.55)^18 (0.45)^(12).
- In R syntax that is dbinom(x = 18, size = 30, prob = 0.55).

3. Calculations (intermediate steps)
- C(30,18) = C(30,12) = 86,493,225.
- (0.55)^18 ≈ 2.120940136 × 10^−5.
- (0.45)^12 ≈ 6.895252355 × 10^−5.
- Multiply: 86,493,225 × (2.120940136e−5) × (6.895252355e−5) ≈ 0.12649 (approximately).
So P(X = 18) ≈ 0.1265.

4. Why this is correct
- The binomial formula directly models the count of successes in a fixed number of independent Bernoulli trials with constant success probability. dbinom in R implements that PMF and returns the exact probability of observing exactly 18 quacks.

5. Why the other options are incorrect
- pnorm(q=18, mean=55, sd=0.5): pnorm is a normal CDF; the parameters are nonsensical here (mean = 55 is not the expected number of successes for n = 30) and we need a discrete binomial probability, not a normal CDF.
- dnorm(x=18, p=0.55, mean=30): dnorm is a normal density function; arguments are wrong (p is not an argument) and the normal is continuous, not the correct discrete model.
- pbinom(q=18, size=55, prob=0.5): pbinom gives a cumulative binomial probability (P(X ≤ q)), not the exact probability P(X = 18), and it uses wrong n and p (size = 55, prob = 0.5) which do not match the experiment.

6. Conclusion
The correct R command is dbinom(x = 18, size = 30, prob = 0.55), which matches the provided answer.

Quackers decides to measure the number of quacks he makes in half an hour. It turns out that it is 18. He wants to calculate the probability that he would have made that many quacks if the underlying true probability of a quack in any minute was 55%. How should he do this?Single choice

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