Teaching a University Course on the Mathematics of Gambling

Courses on the mathematics of gambling have been offered by a number of colleges and universities, and for a number of reasons. In the past 15 years, at least seven potential textbooks for such a course have been published. In this article we objectively compare these books for their probability content, their gambling content, and their mathematical level, to see which ones might be most suitable, depending on student interests and abilities. This is not a book review (e.g., none of the books is recommended over others) but rather an essay offering advice about which topics to include in a course on the mathematics of gambling

Iterating bonferroni bounds

A general method is presented for generating improved Bonferroni bounds whenever an improvement on Booles inequality holds.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/25646/1/0000198.pd

Quantifying HIV transmission flow between high-prevalence hotspots and surrounding communities: a population-based study in Rakai, Uganda.

BACKGROUND: International and global organisations advocate targeting interventions to areas of high HIV prevalence (ie, hotspots). To better understand the potential benefits of geo-targeted control, we assessed the extent to which HIV hotspots along Lake Victoria sustain transmission in neighbouring populations in south-central Uganda. METHODS: We did a population-based survey in Rakai, Uganda, using data from the Rakai Community Cohort Study. The study surveyed all individuals aged 15-49 years in four high-prevalence Lake Victoria fishing communities and 36 neighbouring inland communities. Viral RNA was deep sequenced from participants infected with HIV who were antiretroviral therapy-naive during the observation period. Phylogenetic analysis was used to infer partial HIV transmission networks, including direction of transmission. Reconstructed networks were interpreted through data for current residence and migration history. HIV transmission flows within and between high-prevalence and low-prevalence areas were quantified adjusting for incomplete sampling of the population. FINDINGS: Between Aug 10, 2011, and Jan 30, 2015, data were collected for the Rakai Community Cohort Study. 25 882 individuals participated, including an estimated 75·7% of the lakeside population and 16·2% of the inland population in the Rakai region of Uganda. 5142 participants were HIV-positive (2703 [13·7%] in inland and 2439 [40·1%] in fishing communities). 3878 (75·4%) people who were HIV-positive did not report antiretroviral therapy use, of whom 2652 (68·4%) had virus deep-sequenced at sufficient quality for phylogenetic analysis. 446 transmission networks were reconstructed, including 293 linked pairs with inferred direction of transmission. Adjusting for incomplete sampling, an estimated 5·7% (95% credibility interval 4·4-7·3) of transmissions occurred within lakeside areas, 89·2% (86·0-91·8) within inland areas, 1·3% (0·6-2·6) from lakeside to inland areas, and 3·7% (2·3-5·8) from inland to lakeside areas. INTERPRETATION: Cross-community HIV transmissions between Lake Victoria hotspots and surrounding inland populations are infrequent and when they occur, virus more commonly flows into rather than out of hotspots. This result suggests that targeted interventions to these hotspots will not alone control the epidemic in inland populations, where most transmissions occur. Thus, geographical targeting of high prevalence areas might not be effective for broader epidemic control depending on underlying epidemic dynamics. FUNDING: The Bill & Melinda Gates Foundation, the National Institute of Allergy and Infectious Diseases, the National Institute of Mental Health, the National Institute of Child Health and Development, the Division of Intramural Research of the National Institute for Allergy and Infectious Diseases, the World Bank, the Doris Duke Charitable Foundation, the Johns Hopkins University Center for AIDS Research, and the President's Emergency Plan for AIDS Relief through the Centers for Disease Control and Prevention

The sampling theory of neutral alleles and an urn model in population genetics

The behaviour of a Pólya-like urn which generates Ewens' sampling formula in population genetics is investigated. Connections are made with work of Watterson and Kingman and to the Poisson-Dirichlet distribution. The order in which novel types occur in the urn is shown to parallel the age distribution of the infinitely many alleles diffusion model and consequences of this property are explored. Finally the urn process is related to Kingman's coalescent with mutation to provide a rigorous basis for this parallel.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46946/1/285_2004_Article_BF00276386.pd

Quantifying HIV transmission flow between high-prevalence hotspots and surrounding communities: a population-based study in Rakai, Uganda

Background International and global organisations advocate targeting interventions to areas of high HIV prevalence (ie, hotspots). To better understand the potential benefits of geo-targeted control, we assessed the extent to which HIV hotspots along Lake Victoria sustain transmission in neighbouring populations in south-central Uganda. Methods We did a population-based survey in Rakai, Uganda, using data from the Rakai Community Cohort Study. The study surveyed all individuals aged 15–49 years in four high-prevalence Lake Victoria fishing communities and 36 neighbouring inland communities. Viral RNA was deep sequenced from participants infected with HIV who were antiretroviral therapy-naive during the observation period. Phylogenetic analysis was used to infer partial HIV transmission networks, including direction of transmission. Reconstructed networks were interpreted through data for current residence and migration history. HIV transmission flows within and between high-prevalence and low-prevalence areas were quantified adjusting for incomplete sampling of the population. Findings Between Aug 10, 2011, and Jan 30, 2015, data were collected for the Rakai Community Cohort Study. 25 882 individuals participated, including an estimated 75·7% of the lakeside population and 16·2% of the inland population in the Rakai region of Uganda. 5142 participants were HIV-positive (2703 [13·7%] in inland and 2439 [40·1%] in fishing communities). 3878 (75·4%) people who were HIV-positive did not report antiretroviral therapy use, of whom 2652 (68·4%) had virus deep-sequenced at sufficient quality for phylogenetic analysis. 446 transmission networks were reconstructed, including 293 linked pairs with inferred direction of transmission. Adjusting for incomplete sampling, an estimated 5·7% (95% credibility interval 4·4–7·3) of transmissions occurred within lakeside areas, 89·2% (86·0–91·8) within inland areas, 1·3% (0·6–2·6) from lakeside to inland areas, and 3·7% (2·3–5·8) from inland to lakeside areas. Interpretation Cross-community HIV transmissions between Lake Victoria hotspots and surrounding inland populations are infrequent and when they occur, virus more commonly flows into rather than out of hotspots. This result suggests that targeted interventions to these hotspots will not alone control the epidemic in inland populations, where most transmissions occur. Thus, geographical targeting of high prevalence areas might not be effective for broader epidemic control depending on underlying epidemic dynamics. Funding The Bill & Melinda Gates Foundation, the National Institute of Allergy and Infectious Diseases, the National Institute of Mental Health, the National Institute of Child Health and Development, the Division of Intramural Research of the National Institute for Allergy and Infectious Diseases, the World Bank, the Doris Duke Charitable Foundation, the Johns Hopkins University Center for AIDS Research, and the President's Emergency Plan for AIDS Relief through the Centers for Disease Control and Prevention

The effect of redundancy on probability bounds

AbstractLower bounds on the probability of a union obtained by applying optimal bounds to subsets of events can provide excellent bounds. Comparisons are made with bounds obtained by linear programming and in the cases considered, the best bound is obtained with a subset that contains no redundant events contributing to the union. It is shown that redundant events may increase or decrease the value of a lower bound but surprisingly even removal of a non-redundant event can increase the bound

Pólya-like urns and the Ewens' sampling formula

A Markov process of partitions of the natural numbers is constructed by defining a Pólya-like urn model. The marginal distributions of this process are the Ewens' sampling distributions of population genetics.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46944/1/285_2004_Article_BF00275863.pd

Iterating bonferroni bounds

A general method is presented for generating improved Bonferroni bounds whenever an improvement on Boole's inequality holds.Bonferroni inequalities