John calls a dog by the wrong name 30% of the time. Paul calls a dog...

John calls a dog by the wrong name 30% of the time. Paul calls a dog by the wrong name 10% of the time. They have two dogs Larry and Max. Paul and John are sitting on the couch watching TV and one of the dogs will not stop barking. Paul thinks the dog barking is Max, but John thinks it is Larry. So, they both call out the name they believe to be correct. The dog responds and stops barking. When Max is called by the wrong name, he will respond 40% of the time. When Max is called by the right name, he will respond 90% of the time. Using this, what is the probability that the dog is Max?

Expert Solution

For the dog to be Max, we need to add the probability of John called the wrong name and Max responded at the wrong name and the probability of Paul calling the right name and Max responding to the right name.

Probability of John called the wrong name and Max responded at the wrong name =

Probability of John calling the wrong name * Probability of Max responding to the wrong name

= 0.3 * 0.4

= 0.12

Probability of Paul calling the right name and Max responding to the right name =

Probability of Paul calling the correct name * Probability of Max responding to the correct name

= (1-0.1) * 0.9

= 0.9*0.9

= 0.81.

So, the probability that the dog was Max =

= 0.81 + 0.12

= 0.93.

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orchestra answered 2 years ago

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