Applied statistics for scientists and engineers: Probability- Independence

Question

A musician is currently producing three different songs. Let $A_i$ denote the event that song $i$ is completed by the end of the week, where $i\in\{1,2,3\}$. It is known these three events are independent with probabilities $P(A_1) = 0.4$, $P(A_2) = 0.3$, $P(A_3) = 0.1$. What is the probability that at most one of the three songs is completed by the end of the week?

This is what I thought, but I got it wrong: $$P(A_1\cap A_2\cap A_3)=P(A_1)P(A_2)P(A_3)=0.012$$ The answer given was $0.456$ I don't know how else to do this problem.

*I understand how to do this now, I'm trying to figure out if it asks for exactly one. I thought it was what was P((A_1 n A_2'n A_3')U(A_1'n A_2n A_3')U(A_1'n A_2'n A_3)) But I think that works for "at most one" not "exactly one".

Answer 1

Since its at most one of the three songs :-

P(Answer)= Probability of song 1 alone being published + Probability of song 2 alone being published + Probability of song 3 alone being published

     = .4*(1-.3)*(1-.1) + (1-.4)*.3*(1-.1) + (1-.4)*(1-.3)*.1

     = .456

But i agree with the @ConMan the should be another event where no one publishes a song this is how you get your answer

I think the question should be only one(exactly one) of the three songs published

Answer 2

You're trying to find the probability that at most one song is completed, which is the same as finding the probability that exactly zero or one song is completed. Which means you're looking for two things:

1. The probability that zero songs are completed. (Which is the event $A_1^c \cap A_2^c \cap A_3^c$, where $X^c$ is the complement of event $X$, i.e. it's the event "$X$ doesn't happen".)

2. The probability that one song is completed but the other two aren't. So it's the combination of three events: the event that only $A_1$ is completed, i.e. $A_1 \cup A_2^c \cup A_3^c$, the event that only $A_2$ is completed, i.e. $A_1^c \cup A_2 \cup A_3^c$, and the event that only $A_3$ is completed, i.e. $A_1^c \cup A_2^c \cup A_3$.

The event that you calculated the probability of, $A_1 \cup A_2 \cup A_3$, is the event of "$A_1$ is completed and $A_2$ is completed and $A_3$ is completed", i.e. the event of all three songs being completed.

You should be able to calculate the probabilities of the 4 events I've outlined, noting that all of them are mutually exclusive (meaning that once you've found the separate probabilities, the probability of any of them occurring is their sum).