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Pharmaceutical companies advertise for the birth control pill an annual efficacy of 99.2% in preventing pregnancy. However, under typical use the real efficacy is only about 95%. That is, 5% of women taking the pill for a year will experience an unplanned pregnancy that year. The difference between these two rates is that the real world is not perfect: for example, a woman might get sick or forget to take the pill one day, or she might be prescribed antibiotics which interfere with hormonal metabolism. If a sexually active woman takes the pill for the four years she is in college, what is the chance that she will become pregnant at least once? (Assume that the chance of pregnancy for each year is independent).

(a) Give your answer using the theoretical efficacy of the pill. Please use 4 decimal places. Incorrect: Your answer is incorrect.

(b) Give your answer using the real efficacy of the pill. Please use 4 decimal places

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This looks like a question passed on to us stenographically, including the grammatically imperative mood in parts (a) and (b). Can you ask us a question of your own about this, as opposed to a question written by someone else, which you copied here? –  Michael Hardy Sep 11 '12 at 1:32
@MichaelHardy Why should it matter how she got the question if this is the question she wants help with? –  Michael Chernick Sep 11 '12 at 1:36
you don't have to give me the actual answer. I just need to know how to do it. –  Stephanie Sep 11 '12 at 1:39
@MichaelHardy: This comment of yours is a bit unexpected given this and this and, particularly, this! –  cardinal Sep 11 '12 at 2:29
@MichaelHardy: All three links were intended; I suspect you should have no real difficulties making the connections. Cheers. :) –  cardinal Sep 11 '12 at 14:11

1 Answer 1

(b) The probability of not getting pregnant in any particular year is $0.95$. We first find the probability of not getting pregnant for $4$ consecutive years.

In a sense we are tossing a very biased coin $4$ times, with probability of head equal to $0.95$. We want the probability of $4$ heads in a row. This is $(0.95)^4$. So the probability of not getting pregnant in the $4$ year period is $1-(0.95)^4$.

The calculator gives $(0.95)^4\approx 0.81450625$. To $4$ decimal places, we get $0.8145$. One may want to convert this to "percent" notation. That gives probability $81.45\%$ of not getting pregnant. This kind of precision is mildly, or not so mildly, ridiculous, since undoubtedly $95\%$ is only a rough approximation of the truth. Also, the $95\%$ is presumably obtained from general population data, so may not apply to a restricted subpopulation, such as college students.

Finally, the probability of getting pregnant at least once is approximately $1-0.8145$, that is, $0.1855$, or in percent, $18.55\%$.

Problem (a) is done in the same way. Just as a check on the correctness of your calculation, I get that the probability of getting pregnant in $4$ years is about $0.031618043$.

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And, presumably, having seen how to do (b), OP can now see how to do (a). –  Gerry Myerson Sep 11 '12 at 2:39

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