Conditional Probability and Independence

A sample space is generally defined and all probabilities are calculated with respect to that sample space. In many cases, however, we ate in a position to update the sample space based on new information. For example, like the fourth example of Example 2.3, if we just consider the case that two outcomes from roUing a die twice are the same, the size of the sample space is reduced from 36 to 6. General definitions of conditional probability and independence are introduced. The Bayes theorem is also introduced, which is the basis of a statistical methodology called Bayesian statistics. 

Consider two events, A and B. Suppose that an event B has occurred. This occurrence may change the probability of A. We denote this by P A B), the conditional probability of event A given that B has occurred. 

As an example, suppose a card is randomly dealt from a well-shuffled deck of 52 cards. The probability of this card being a heart is since there are 13 hearts in a deck. However, if you arc told that the dealt card is red, then the probability of a heart changes to because the sample space is reduced to just 26 cards. Likewise, if you are told that the dealt card is black, then P(Heart Black) = 0. 

Example 2.4 Conditional Probability from Tossing Two Dice. An experiment consists of tossing two fair dice with a sample space of 6 x 6 = 36 outcomes. Consider two events  

Thus B and A D B consist of 12 and 2 outcomes, respectively. Assuming that all outcomes are equally likely, the conditional probability of A given B is 

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