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More ways to get Braingle...

Let's Go Out to the Movies

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

 

Puzzle ID:#31308
Fun:*** (2.46)
Difficulty:*** (2.85)
Category:Probability
Submitted By:tsimkinAus*****!
Corrected By:Zag24

 

 

 



Gretchen and Henry were sent to their rooms for fighting in the house. They each separately voiced their protest to their father, insisting that the fight was nothing more than healthy sibling competition, and they each wanted to go out that afternoon to see a movie. He was moved by their stories, but wouldn't simply set them free. Instead, he devised a system. He went to each child's room with a penny, and told them that they would have to show up in the den in 10 minutes, and could choose to bring the penny with them or leave it in their respective rooms. Dad would then flip the one or two pennies brought to the den, and if the pennies he flipped came up heads, the kids could go to the movies. If neither brought a penny, or if he flipped at least one tail, they would stay in their rooms until supper time.

The problem facing Gretchen and Henry was that neither knew what the other would do. It would be easy if they could collude -- one would bring a penny, and the other would not, giving them a 50% chance of going free -- but they did not have this luxury.

If they both acted optimally, what is the probability that they will be free in time to see the movie?


Hint

If they both bring a penny 100% of the time, they will never be sent back without a flip, but they will only have a 25% chance of going free. Can they do better? Acting optimally means they will have to pursue some course of action some percentage of the time.
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