Question

Savannah has entered a dart-throwing competition. Each time she throws a dart, she hits the bullseye with probability p=0.15, independent of any other throw. She throws the dart until she hits the bullseye, at which point her round is over. Let D denote the number of times the dart is thrown.

a. What is the PMF PD(d) ?

Answer 1

The probability mass function (PMF) PD(d) for the number of darts thrown until Savannah hits the bullseye is given by PD(d) = (1-p)^(d-1) × p, with p = 0.15 representing the probability of hitting the bullseye on any given throw.

Step-by-step explanation:

The question deals with finding the probability mass function (PMF) PD(d) for the number of darts thrown until Savannah hits the bullseye. Since Savannah hits the bullseye with a probability p = 0.15 on any throw, and each throw is independent, the scenario can be modeled as a geometric distribution. The PMF of a geometric distribution is given by PD(d) = (1-p)^(d-1) × p, where d is the number of trials until the first success (hit).

For example, if we want to find the probability that Savannah hits the bullseye on her third dart throw, we would calculate PD(3) = (1-0.15)^(3-1) × 0.15 = (0.85)^2 × 0.15.