Am I thinking of this the right way?

Hi, I'm not sure if I'm thinking of the right way to do this problem:

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!
The way I understand the problem statement the percents are referring to the total -- 35% of the total are bacilli and 11% of the total are gram-positive bacilli. So trying to figure out what % bacilli are not gram-positive is as simple as taking the total bacilli (35%) minus the gram-positive bacilli (11%) = 24%. For #3, we have to count the bacilli and the gram-positive, which is naively 35% + 23% = 58%. But 11% is counted twice (in both bacilli and gram-positive counts) so that needs to be removed: 58% - 11% = 47%. This is just an arithmetic way of doing what rogojel suggested, using Venn diagrams.