Consider our previous experiment of tossing a coin twice and recording the outcome after each toss..
Consider our previous experiment of tossing a coin twice and recording the outcome after each toss. A valid probability measure P on the sample space D fHH; HT; TH;T T g of this experiment is one that assigns probability 1 4 to each of the four sample points; i.e., P .HH/ D P .HT/ D P .TH/ D P .T T / D 1 4 . By the additivity property assumed in the definition, if we consider the event A D fHT;THg = the statement that exactly one head and exactly one tail will be obtained, then P .A/ D P .H T / C P .TH / D 1 4 C 1 4 D 1 2 . If we believed in this probability, then a bet that offers to pay us ten dollars should the event A happen and require us to pay ten dollars if it does not happen would be considered a fair bet. Indeed, the original development of probability was motivated by betting and gambling scenarios involving coin, dice, or card games. Because of this, and also because they seem to provide an endless supply of interesting problems and questions, many of our examples will be based on suitable coin, dice, and card experiments. Apr 06 2022 11:33 AM
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