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Compute bus-on-time and dice-sum probabilities

SIG (Susquehanna)Data Scientist · HR Screen
MediumData ScientistHR ScreenCoding & Algorithms

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This problem tests your ability to calculate probabilities involving continuous uniform distributions and discrete combinations of independent events.

Probability Problems

Part 1: The Bus Problem

A shuttle bus arrives at a stop at a random time uniformly distributed between 8:00 AM and 8:30 AM. You arrive at the bus stop at 8:10 AM. What is the probability that you wait for less than 5 minutes?

Part 2: The Dice Problem

You roll two fair, six-sided dice. What is the probability that the sum of the two dice is a prime number? Provide your answer as a simplified fraction.

Requirements

For the bus problem, define the arrival time as a continuous random variable T. Show the logic used to determine the favorable window for T relative to your arrival time. For the dice problem, you may use a counting method or a probability table.

Examples

Bus Example:

  • If the bus arrives at 8:12 AM, your wait time is 2 minutes (success).
  • If the bus arrives at 8:20 AM, your wait time is 10 minutes (failure).

Dice Example:

  • If one die is 1 and the other is 1, the sum is 2 (prime).
  • If one die is 6 and the other is 6, the sum is 12 (not prime).

Hints

Hint 1

For the bus problem, identify the interval of bus arrival times that result in a wait of less than 5 minutes given you are already at the stop.

Hint 2

For the dice problem, list all possible sums from 2 to 12 and identify which of those values are prime numbers.

Hint 3

Remember that the total number of outcomes for two dice is 6 * 6 = 36, and each pair is equally likely.

Follow-up questions

  • How would the bus probability change if you arrived exactly at 8:00 AM versus 8:25 AM?
  • If you rolled three dice instead of two, how would you efficiently count the number of outcomes that sum to a prime number?
  • How would you generalize the bus problem if the bus arrival followed an exponential distribution instead of a uniform distribution?