$A$, $B$, and $C$ have $10\%$, $20\%$ and $30\%$ chance of independently solving a certain math problem. If they all try independently of one another, what is the probability that this group will solve the problem?
Isn't the solution just $0.1 * 0.2 * 0.3$? I don't get why the worksheet says the answer is $1 – (0.9 * 0.8 * 0.7)$. It doesn't make sense to me.
Best Answer
$0.1*0.2*0.3$ is the probability that A solves the problem and B solves the problem and C also solves the problem. The question is asking for the probability that the group solves the problem and in order for that to happen only one of A, B, or C needs to solve the problem; in other words, the probability that at least one of them solves the problem.
The probability of at least one of them solving it is the complement of the probability that none of them solves it; in general, $P(\text{at least one}) = 1 - P(\text{none})$.
$0.9 * 0.8 * 0.7$ is the probability that none of them solves it (the probability that A does not solve the problem and B does not solve the problem and neither does C). Take the complement and you have your solution:
$$P(\text{A or B or C solves the problem}) = 1 - (0.9 * 0.8 * 0.7) $$