Question

A group of

220

patients tested a new medication.
Some tried the new medication, and the rest took the old medication.

The results are reported in the following two-way frequency table.

	Improvement	No improvement
New medication	66	22
Old medication	55	77

A patient is chosen at random from this group.

Complete the following. Write your answers as decimals.

(a)Find the probability that the patient showed improvement.

=P(improvement)

(b)Find the probability that the patient showed improvement, given that he took the new medication.

=P(improvement | new medication)

(c)Is there evidence that a patient who takes the new medication is more likely to show improvement than a randomly chosen patient from the group?

Yes, because the probability found in part (b) is much greater than the probability found in part (a).

No, because the probability found in part (b) is much greater than the probability found in part (a).

Yes, because the probability found in part (a) is much greater than the probability found in part (b).

No, because the probability found in part (a) is much greater than the probability found in part (b).

Yes, because the probability found in part (b) is about the same as the probability found in part (a).

No, because the probability found in part (b) is about the same as the probability found in part (a).

Answer 1

a) The probability of showing improvement is computed here as:
P( improvement )

= n(improvement) / n(Total)

= (66 + 55) / 220

= 0.55

Therefore 0.55 is the required probability here.

b) Now the probability that the patient showed improvement, given that he took the new medication is computed using Bayes theorem as:
P( improvement | new medication) = n(improvement and new medication) / n(new medication)

= 66 / (66 + 22)

= 3/4 = 0.75

Therefore 0.75 is the required probability here.

c) As the probability of showing improvement after medication is 0.75 > probability of showing improvement that is 0.55, therefore Yes, because the probability found in part (b) is much greater than the probability found in part (a) is the correct answer here.