A space probe near Neptune communicates with Earth using bit...

A space probe near Neptune communicates with Earth using bit strings. Suppose that in its transmissions it sends a 1 one-third of the time and a 0 two-thirds of the time. When a 0 is sent, the probability that it is received correctly is 0.9, and the probability that it is received incorrectly (as a 1) is 0.1. When a 1 is sent, the probability that it is received correctly is 0.8, and the probability that it is received incorrectly (as a 0) is 0.2. a) Find the probability that a 0 is received. b) Use Bayes’ theorem to find the probability that a 0 was transmitted, given that a 0 was received.

A space probe near Neptune communicates with Earth using bit strings. Suppose that in its transmissions it sends a 1 one-third of the time and a 0 two-thirds of the time. When a 0 is sent, the probability that it is received correctly is 0.9, and the probability that it is received incorrectly (as a 1) is 0.1. When a 1 is sent, the probability that it is received correctly is 0.8, and the probability that it is received incorrectly (as a 0) is 0.2.
a) Find the probability that a 0 is received.
b) Use Bayes’ theorem to find the probability that a 0 was transmitted, given that a 0 was received.

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Key Concepts

Law of Total Probability

The Law of Total Probability allows us to compute the probability of an event by considering all the possible ways the event can occur through a systematic partitioning of the sample space. It involves summing the weighted probabilities over all mutually exclusive scenarios, which is essential in many probabilistic and statistical methods.

Bayes’ Theorem

Bayes’ Theorem is a method for updating the probabilities of hypotheses when given new evidence or data. It connects conditional probabilities with their inverses and is widely utilized to revise prior beliefs or predictions, particularly in contexts where uncertainty or error is present.

Conditional Probability

This concept measures the likelihood of an event occurring given that another event has occurred. It forms the basis for understanding how new information can affect the probability of outcomes and is essential for reasoning about situations in which events influence each other.

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