Question

The following data describe a clinical trial of an allergy medication:

Allergy Drug Placebo Control Total

Improvement 65 42 31 138

No Improvement 55 58 49 162

Total 120 100 80 300

a. What is the probability that a randomly selected person in the study was given either the drug or the placebo?

b. What is the probability that a randomly selected person either improved or did not improve?

c. What is the probability that a randomly selected person either was given the drug or improved?

d. What is the probability that a randomly selected person was given the drug and improved?

Answer 1

Given Data is

Allergy	Drug	placebo	control	Total
improvement	65	42	31	138
NO Improvement	55	58	49	162
Total	120	100	80	300

Probability of an event = Number of times the event occurred / total number

a. What is the probability that a randomly selected person in the study was given either the drug or the placebo?

P(randomly selected person in the study was given either the drug or the placebo) = P(randomly selected person in the study was given the drug + P(randomly selected person in the study was given the placebo) - P(randomly selected person in the study was given either the drug AND the placebo)

P(randomly selected person in the study was given either the drug or the placebo) = ( 100/300 ) + (80 / 300) - 0/300 = 0.6

So , there is probability of 0.60 that a randomly selected person was given either the drug or the placebo.

b. What is the probability that a randomly selected person either improved or did not improve?

P(a randomly selected person either improved or did not improve) = P(a randomly selected person improved ) + P(a randomly selected person did not improve) - P(a randomly selected person either improved AND did not improve) = (138/300) + (162/300) - (0/300)= 1

Note since there are only two categories in this are either improved or not improved , thus the probability will have to be 1 as there is no other valid possibility other than a person improved or not improved

So , there is probability of 100% that a randomly selected person was either improved or not improved

c. What is the probability that a randomly selected person either was given the drug or improved?

P(a randomly selected person either was given the drug or improved) = P(that a randomly selected person either was given the drug) + P(that a randomly selected person either was improved) - P(a randomly selected person either was given the drug AND improved)

P(a randomly selected person either was given the drug or improved) = (120/300) + (138/300) - 65/300 = 0.6433

So , there is probability of 0.6433 that a randomly selected person either was given the drug or improved

d. What is the probability that a randomly selected person was given the drug and improved?

P( a randomly selected person was given the drug and improved ) = number of persons who were given drug AND they improved / Total Number of persons = 65/300

P( a randomly selected person was given the drug and improved ) = 0.216667

So , there is probability of 0.2166667 that a randomly selected person was given the drug and improved

ADDITIONAL EXPLANATION -

In the case of P( A or B ) , we will have to use the set theory that P(A or B ) = P( A union B ) = P(A) - P(B ) + P(A and B) as the common part in A and B was included twice , once while calculating P(A) and the other time by calculating P(B) , so it has to be deducted once to obtain the correct answer.