There are three caskets of treasure. The first casket contains 3 gold coins, the second casket contains 2 gold coins and 2 bronze coins, and the third casket contains 2 gold coins and 1 silver coin. You choose one casket at random and draw a coin from it. The probability that the coin you drew is gold has the value ab, where a and b are coprime positive integers. What is the value of a+b?
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probability has to be <1. It can't be a product of two integers.
I drew a probability tree. Such a tree follows a process thru its various stages with probabilities being calculated at each stage. From the starting point, the probability of selecting each one of the three caskets is 1/3. If the first casket is selected then each of the three gold coins has a 1/3 probability of being selected. The selection of a casket and the selection of a coin are separate and independent sub-events. The multiplication theorem of probability states that if a compound event (such as the selection of a particular gold coin from the first basket) is made up of a number of separate and independent sub-events, the probability of the compound event is the product of the separate probabilities of each sub-event.
The addition theorem of probability states that if an event (such as selecting a gold coin from any one of the three caskets) can happen in a number of different and mutually exclusive ways, the probability of its happening at all is the sum of the separate probabilities that each event happens.