The Addition Rule Lecture 44 Section 9.3 Robb T. Koether Hampden-Sydney College Mon, Apr 4, 204 Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 / 7
The Addition Rule 2 3 Assignment Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 2 / 7
Outline The Addition Rule 2 3 Assignment Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 3 / 7
The Addition Rule Theorem Suppose a finite set A is the union of k mutually disjoint sets A, A 2,..., A k. Then A = A + A 2 + + A k. A consequence of this is that if an event E is the union of k mutually disjoint events E, E 2,..., E k. Then P(E) = P(E ) + P(E 2 ) + + P(E k ). Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 4 / 7
Examples A card is drawn from a deck. What is the probability that it is a spade or a heart? What is the probability that it is a 5, an 8, or a face card? Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 5 / 7
The Difference Rule Theorem Let A be a finite set and B A. Then A B = A B. A consequence of this is that if E and E 2 are events and E 2 E, then P(E E 2 ) = P(E ) P(E 2 ). Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 6 / 7
Examples A card is drawn from a deck. What is the probability that it is a face card, but not a Jack? What is the probability that it is red, but not a Heart? A C++ identifier must begin with a letter or an underscore. That may be followed by any number of letters, digits, and underscores. How many distinct C++ identifiers of length at most 8? Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 7 / 7
Outline The Addition Rule 2 3 Assignment Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 8 / 7
Theorem Let A and B be finite sets. Then A B = A + B A B. A consequence of this is that if an E and E 2 are events, then P(E E 2 ) = P(E ) + P(E 2 ) P(E E 2 ), or P(E or E 2 ) = P(E ) + P(E 2 ) P(E and E 2 ) Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 9 / 7
A and B Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 0 / 7
A Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 0 / 7
2 A + B Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 0 / 7
A + B A B Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 0 / 7
This rule can be applied recursively to derive the Inclusion/Exclusion Rule for 3 or more sets. Let A, B, and C be finite sets. Then A B C = A (B C) = A + B C A (B C) = A + B + C B C (A B) (A C) = A + B + C B C ( A B + A C (A B) (A C) ) = A + B + C B C A B A C + A B C. Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 / 7
A and B Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 2 / 7
A Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 2 / 7
2 2 A + B Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 2 / 7
2 2 3 2 A + B + C Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 2 / 7
2 2 2 A + B + C A B Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 2 / 7
2 A + B + C A B A C Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 2 / 7
0 A + B + C A B A C B C Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 2 / 7
A + B + C A B A C B C + A B C Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 2 / 7
What would the Inclusion/Exclusion Rule be for 4 sets? Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 3 / 7
Example Suppose that the Math/CS department has a total of 5 majors. 2 are majoring in Math. 4 are majoring in Applied Math. 4 are majoring in Computer Science. 3 are majoring in Math or Applied Math. 6 are majoring in Applied Math or Computer Science. 2 are majoring in only Computer Science. is majoring in Math and Computer Science. How many students are majoring in all three? Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 4 / 7
Outline The Addition Rule 2 3 Assignment Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 5 / 7
Collected Collected Sec. 8.5: 23, 50. Sec. 9.: 2, 20. Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 6 / 7
Assignment Assignment Read Sections 9.3, pages 540-549. Exercises 6, 7,, 2, 2, 22, 25, 27, 33, 34, 42, page 549. Robb T. Koether (Hampden-Sydney College) The Addition Rule Mon, Apr 4, 204 7 / 7