Bacterial cells are often classified by their shapes, the three most common of these are cocci (spherical), bacilli (rod shaped), and spirilla (spiral). Another useful method for classification is whether the bacteria’s cell walls turn purple when exposed to a gram stain (gram positive). A researcher collected a sample of water from a nearby stream, the sample was found to have 2,373 distinct bacterial cells. 35% of the cells were bacilli, 23% were gram positive, and 11% were gram positive bacilli.

1. Are being bacilli and gram positive disjoint, i.e. mutually exclusive?

2. What percent of cells are bacilli but not gram positive?

3. What percent of cells are bacilli or gram positive?

4. What percent of cells are neither bacilli nor gram positive?

5. Is the event that a cell is a bacilli independent of the event that the cell is gram positive

For number two, I thought of it like P(bacilli and not gram positive), so I multiplied .35 and (1-.23) to get .27. For 3, since the events are not disjoint, I did (.35 + .23) - (.35)(.23). For 4, I simply found the complement of gram positive bacteria, 1-.11 = .89. Am I going about these the right way? Thanks for your help!