Algorithms Q&A - Recent questions in Probability

Coin - Real or Biased

Tue, 25 Apr 2023 17:40:07 +0000

There are two coins: Fair Coin: Heads and Tails with 50% Probability, and Biased Coin: Always lands on Heads. Dealer uses either coin with equal probability.

We see that the dealer tosses the coin, and it comes in a sequence as H, H, T. We can immediately infer that the dealer is using the fair coin (because biased coin never lands on Tails).

In next run, the dealer tosses the coin, and it comes in a sequence as H, H, H. What is the probability that the dealer is using the biased coin.

The dealer does not switch coins during a single run.

Disco Fever and Two Kinds of Tests

Mon, 10 May 2021 00:52:23 +0000

25% of the people in a city have Disco Fever. There are two tests for Disco Fever, Rapid Antigen Test (RAT) and Full Scan Test (FST). When a person goes to a doctor to test for Disco Fever, with probability 2/3 the doctor prescribes RAT, and with probability 1/3 the doctor prescribes FST.

When RAT is done on a person, the outcome is as follows: If the person has the disease, the result is positive with probability 3/4. If the person does not have the disease, the result is positive with probability 1/4.

When FST is done on a person, the outcome is as follows: If the person has the disease, the result is positive with probability 1 (that is, FST has no false negatives). If the person does not have the disease, the result is positive with probability 1/3.

A person is picked uniformly at random from the town and is sent to a doctor to test for Disco Fever. The result comes out positive. What is the probability that the person has Disco Fever?

Probability of rain in desert when a reliable weatherperson predicts one

Sun, 21 Feb 2021 18:38:53 +0000

Antonio has an exciting soccer game coming up. In recent years, it has rained only 5 days each year in the city where they live.

Unfortunately, the weatherperson has predicted rain for that day. When it actually rains, she correctly forecasts rain 90% of the time. When it doesn't rain, she incorrectly forecasts rain only 10% of the time.

What is the probability that it will rain on the day of Antonio’s soccer game?