15.Math-Review. Thursday 8/17/00. Event A, and event A c , the complement of A:. U. A. A c. Venn Diagrams. Notation: The complete set:. U. A B. A. B. A B. B included in A, or B A The null set is labeled . Venn Diagrams. Notation: A and B, called A B A or B, called AB.
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1. A B = B A
2. A ( B C ) = ( A B ) C
3. A ( B C ) = ( A B ) ( A C )
4. ( A c) c = A
5. ( A B )c = Ac Bc
6. A Ac =
7. A U = A
0 P(A) 1
P( A B ) = P(A) + P(B)
a. both systems fail in the next hour
b. the primary system does not fail over the next hour
c. the primary system fails in the next hour but not the backup
d. exactly one of the two systems fails in the next hour
e. at least one system fails in the next hour
Suppose p1 p2. Under this modification, find the chance that:
f. both systems fail in the next hour
g. exactly one of the systems fails
h. at least one of the systems fails
a. her first two purchases yield a J and R, in that order
b. she is eligible for a free pizza after her first two purchases
c. it takes her exactly three purchases to achieve the bonus
d. even after buying five pizzas, she is not eligible for her first complimentary one.
a. If Mendel had invested in four recently-patented products that he chose at random, what is the probability that all four were commercially successful?
b. Given that all four of his products were commercially successful, what is the chance that they all involved biotechnology?
c. Let Q be the fraction of recently patented nonbiotechnological products that were successful. Given the information above, what is the numerical value of Q? (HINT: Express the overall success rate for recently-patented products in terms of Q.)
d. Are the successfulness and biotechnicality of a project independent events?
P(AN|A1 and A2 and ... AN-1) x P(A1 and A2 and ... AN-1)
P(Ai|A1 and A2 and ... Ai-1) = [365 - (i - 1)]/365 = (366 - i)/365
Example: The number on top after the roll of a die can be 1, 2, 3, 4, 5 or 6.
Example: If we draw a student at random in the class and record their height in principle we can obtain any number between .5 meters and 2.5 meters. (Very unlikely in the extremes)