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Condition on a Chosen Rabbit Losing

SIG (Susquehanna)Data Scientist · Online Assessment
EasyData ScientistOnline AssessmentStatistics & Math

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Calculate the conditional probability that a randomly selected rabbit is the least athletic of three rabbits given that it lost its benchmark race, using Bayes' theorem.

Problem Description

A racing stable trains three rabbits of differing athletic abilities:

  • Rabbit A (the most athletic)
  • Rabbit B (moderately athletic)
  • Rabbit C (the least athletic)

Each rabbit is evaluated by running against a fixed course standard. Because their athletic conditioning differs, they do not have equal race outcomes:

  • Rabbit A fails to beat the standard (loses) with probability 1/4.
  • Rabbit B fails to beat the standard (loses) with probability 1/3.
  • Rabbit C fails to beat the standard (loses) with probability 1/2.

A rabbit is chosen uniformly at random from the three and runs the standard trial. You are informed that the chosen rabbit lost.

Task

  1. Formulate the prior probabilities and conditional likelihoods for each rabbit.
  2. Compute the total probability that a randomly chosen rabbit loses.
  3. Derive the posterior probability that the chosen rabbit was the least athletic rabbit (Rabbit C), expressing your final answer as an irreducible fraction.

Hints

Hint 1

Identify the prior probability of selecting each rabbit, which is 1/3 for each since the choice is uniformly random.

Hint 2

Use the law of total probability to sum the joint probabilities of selecting each rabbit and having that rabbit lose.

Hint 3

Apply Bayes' rule by taking the joint probability of selecting Rabbit C and it losing, divided by the total probability of a loss.

Follow-up questions

  • How does the posterior probability change if the chosen rabbit runs two independent trials and loses both?
  • If the trainer instead selects a rabbit with probability proportional to its athleticism (e.g., weights 3 : 2 : 1 for A, B, and C), what is the updated posterior probability of Rabbit C given a loss?
  • Suppose the failure probabilities are unknown parameters ordered as p_A < p_B < p_C. What inequality must hold among the parameters for the posterior probability P(C | Lost) to exceed 1/2?