By William Feller

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**Additional resources for An Introduction to Probability Theory and its Applications**

**Example text**

5 39 Independence For the case of three events, AI , A2 , and A3, independence amounts to satisfying the four conditions P (A1 n A2 ) = P (A I ) P (A2 ) , P ( A 1 n A3) = P(At ) P(A3 ) , P (A2 n A3) = P (A2 ) P (A3) , P(A1 n A2 n A3) = P (A1 ) P (A2 ) P (A3 ) . The first three conditions simply assert that any two events are independent, a property known as pairwise independence. But the fourth condition is also important and does not follow from the first three. Conversely, the fourth condition does not imply the first three; see the two examples that follow.

N An - d· to some i n te is con d i tional probab i l ities u p to that node. For exampl e , t he e vent Al to the node s hown i n n figure . and i ts is n r- A2 n A3 Sec. 3 25 Conditional Probability and by using the definition of conditional probability to rewrite the right-hand side above as For the case of just two events, A l and A2 , the multiplication rule is simply the definition of conditional probability. Example 1 . 10. Three cards are drawn from an ordinary 52-card deck without replacement (drawn cards are not placed back in the deck).

And The key is to choose appropriately the problem structure. Here are some exal nples . by theorem . T he Visualization and verification of the total space. so t he event B c a n be events , . . 13: B= I n B) u . . u Usi n g t he a d d it i v i t y ax iom. it follows that P ( B ) = P( A I n B) t he t he d e fi n i t i o n ( A rl + . . + cond itional P ( Ai n B) = B ) = P ( A t )P ( B I A d n n B). ). + . . + BI ). For a n alternat ive view. consider an equivalent seq uential model . as shown on the right .