$5.$ Tamika and Catalina work as plumbers, each serving the same neighborhood. Tamika is a careful worker who utilizes the best replacement parts on her jobs. Catalina worries less about the quality of her jobs. She tries to complete them quickly and often uses low$-$cost replacement parts. Tamika believes that if customers knew the quality of her work, she could charge $$100$ for a typical job. If customers recognized that Catalina$$s work was of lower quality, she would be able to charge only $$60$ for a typical job. Because customers cannot judge the quality of the two plumbers$$ work, they consider it equally likely that the work is high quality $($i$.$e$.,$ a $$100$ job$)$ or low quality $($i$.$e$.,$ a $$60$ job$).$ Thus, both Tamika and Cataline are able to charge $$80$ for a typical job. Recently, Tamika placed an advertisement in the local paper emphasizing the high quality of her work. She was angered and dismayed when Catalina placed a similar advertisement the following week. $(30$ points$)$

a$.$ Describe the problem Tamika faces. Do you expect that Tamika$$s advertisement will help her to better compete with Catalina? Why or why not?