Odds are defined as the number of successes to the number of failures. Let's first look at number of ways to get a woman in the Army. By looking at the graph we can see that there are 10,000 women in the army that we can choose. Next, lets look at how many ways we can fail. So we want to know how many people are there that aren't women in the army. This is 420,000 + 320,000 + 60,000 + 300,000 + 70,000 + 170,000 +80,000 = 1420,000. So the odd are 10000:420000 but notice that this can be simplified to 1:142 by dividing both numbers by 10000. So the odds in favor are 1:142.
The odds against the person is the exact opposite. This time we define a success as getting someone who isn't a women in the army; there are 1420000 cases like this as calculated about. And now the failures are when we do get a woman in the army; there are 10000 of these cases also from above. So we have 1420000:10000, which can be simplified to 142:1. Notice that the odds against are the odds in favor reversed as we just switched what we called a success and failure.