Bayes’ Theorem
1. You are presented with two bags, one containing two ham sandwiches and the other containing a ham sandwich and a cheese sandwich. You reach in one bag and draw out a ham sandwich. What is the probability that the other sandwich in the bag is also a ham sandwich?
2. You are presented with three bags: two contain a chicken-fat sandwich and one contains a cheese sandwich. You are asked to guess which bag contains the cheese sandwich. You do so, and the bag you selected is set aside. (You obviously have one chance in three of guessing correctly.) From the two remaining bags, one containing a chicken-fat sandwich is then removed. You are now given the opportunity to switch your selection to the remaining bag. Will such a switch increase, decrease, or leave unaffected your chances of correctly ending up with the bag with the cheese sandwich in it?
Exercise V
In a fair lottery with a million tickets, the probability that ticket #1 will lose is 0.999999, the probability that ticket #2 will lose is also 0.999999, and so on for each ticket. On this basis, you might seem justified in believing each premise in the following argument:
Ticket #1 will lose. Ticket #2 will lose. Ticket #3 will lose. [and so on, until:] Ticket #1,000,000 will lose. There are only 1,000,000 tickets.
