Suppose that the probability of exposure to the flu during epidemic is 6. Experi

Question

Suppose that the probability of exposure to the flu during epidemic is 6. Experience has shown that a serum is 80% successful in preventing an inoculated person from acquiring the flu, if exposed to it. A person no inoculated faces a probability of 0.90 of acquiring the flu if exposed to it. Two persons, one inoculated and one not, perform a highly specialized task in a business. Assume that they are not at the same location, are not in contact with the same person people and cannot expose each other to the flu. What is the probablility that at least one will get the flu?

Explanation / Answer

The .6 means that there's a 60% chance of being exposed to the flu, and a 40% chance of not. But the problem goes on to say that if you are exposed, then if you've been inoculated there is still a .2 (20%) chance of getting sick (because .8 of the time you'll be protected), while if you haven't been inoculated then there is a 90% chance of getting sick. So for an inoculated person, the chance of getting flu is .6 x .2 = .12 or 12% For a not-inoculated person, the chance of getting flu is .6 x .9 = .54 or 54% Finally, they want to know the chance either (or both) of these two people will get the flu. One way to calculate that is to figure the chance that neither will get flu, and subtract that from 1.0 (100%). The chance of two independent (that's why there's all that talk about their not interacting, encountering the same people, etc.) events both happening is each probability times the other (the product). So the inocuted person will be flu-free 1 - .12 = 88% of the time and the not-inoculated person will be flu-free 1 - .54 = 46% of the time The chance both will be flu-free is .88 x .46 = .4048 (40.48%) The chance one, or the other, or both will catch flu is 1 - .4048 = .5952 (59.52%). [if you want to only work from the chance each will get sick, you have to subtract for the double-counting of the case where both of them get sick: .12 + .54 - (.12 x .54) = .5952 This second way is easier, if you understand why you're subtracting the .12 x .54 ] That's it!

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