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5. Prove that if A1, A2, ... An and B are sets, then (A. – B) U (A2 – B) U... U (An – B) = (A, U A, U... U An) – B.
1. Prove“inclusion-exclusion,”thatP(A∪B)=P(A)+P(B)−P(A∩B). 2. Prove the “unionbound, ”thatP(A1∪A2)≤P(A1)+P(A2). Under what conditions does the equality hold? 3. Provethat, for A1 andA2 disjoint, P(A1∪A2|B)=P(A1|B)+P(A2|B). 4. A and B are independent events with nonzero probability. Prove whether or not A and Bc are independent.
Prove or disprove: for all sets A, B, C and D, (Ax B) U (Cx D) (AUC) x (BUD).
Let A1, A2, ...An Prove : P(Un k=1 Ak) = P9A1) + P(A1c ...... Problem 4.Let A1, A2, . . . , An be events. Prove
P.4 Prove that for any set of integers {ao, aj, a2,..., ax), the integer n=ax. 10% +ax-1·10k-1 + ... + 01.10+ 0 is congruent to E-01–1)' a; (mod 11). What significance does this hold when the ai are restricted to the set {0,1,2,3,4,5,6,7,8,9}?
L: R3 to R3 defined by L([a1 a2 a3]) = [a1 a2^2+a3^2 a3^2]. Prove that this is a linear transformation or not.
Suppose that A1,A2,.., Ak are mutually exclusive events and P(B)>0. Prove that
Matrix notation: A=(a1,a2,a3.....,an) = [a1 a2 a3 a4 .....an] are they equal? look at the sample picture A should be matrix but it uses ( ) rather than [ ] Given that A is an n×n matrix with the property AX = 0 for all X " 1 A=(a,,a,, 0 0 Let a.) Let e, =| | | ← ith element Comment
(a) Prove that if A1, A2, . . . , are mutually exclusive, then P(An) → 0 as n → oo. (Recall that whenever Σοοι pn is finite and all the pn's are nonnegative, then Pn-+0 as n o.) (b) Suppose 1 flip a fair coin forever. Let An be the event that the rnth flip is a head. Since the coin is fair, P(An)-.Notice that P(An) 0 asnoo. How, then, can the previous problem still be true? Each An...
4. Say we have a population, with Ne = 100, containing two alleles, A1 and A2 at frequencies 0.6 and 0.4, respectively. We leave this population alone for 10000 generations (keeping its size constant), then come back to find that the allele frequencies are still 0.6 and 0.4. a) Explain why this would be evidence that selection is acting in this population. b) Which genotype would you expect to have the highest fitness in this case? Explain.