Question 1198140
.
 At the start of flu season, Dr. Anna Ahmeed examines 50 patients over two days. Of those 50 patients, 30 have a 
headache, 24 have a cold, and 12 have neither symptom. Some patients have both symptoms.

a) Draw a Venn diagram

b) Find the probability that a patient selected at random has just a headache. Express the answer in fraction form lowest terms. 
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<pre>
                        <U>Step by step</U>


Step 1.  Find the probability that a randomly selected patient has a headache or a cold

             P(at least one of both symptoms) = 1 - P(neither symptom) = 1 - {{{12/50}}} = {{{38/50}}}.



Step 2.  Find the probability that a randomly selected patient has both symptoms.

         Use the formula

             P(at least one symptom) = P(headache) + P(cold) - P(both symptoms).


         It gives

             P(both symptoms) = P(headache) + P(cold) - P(at least one symptom) = {{{30/50}}} + {{{24/50}}} - {{{38/50}}} = {{{16/50}}}.



Step 3.  The probability that a randomly selected patient has just a headache is

             P(just a headache) = P(at least one symptom) - P(both symptoms) = {{{38/50}}} - {{{16/50}}} = {{{22/50}}}  = {{{11/25}}}.    <U>ANSWER</U>
</pre>

Solved.