Several studies have been conduct on the effectiveness and safety of the H1N1 influenza vaccine. One study found that in a group that received the actual medication, the drug was 87% effective. In a group that received a placebo, there was 10% effective rate. If the study used 400 people to received the placebo and 600 people to get the actual drug, determine the probability that ant person in the study will
1) received the drug and not get sick
2) not get sick.
To calculate the probability that a person in the study will receive the drug and not get sick, we need to multiply the probability of receiving the drug by the effectiveness rate of the drug.
1) Probability of receiving the drug:
Out of the total study population, 600 people received the actual drug.
Probability of receiving the drug = Number of people receiving the drug / Total study population
Probability of receiving the drug = 600 / (600 + 400)
Probability of receiving the drug = 0.6 or 60%
Effective rate of the drug = 87%
Probability of receiving the drug and not getting sick = Probability of receiving the drug x Effective rate of the drug
Probability of receiving the drug and not getting sick = 0.6 x 0.87
Probability of receiving the drug and not getting sick = 0.522 or 52.2%
2) To calculate the probability of not getting sick, we need to consider both the group that received the drug and the group that received the placebo.
Probability of not getting sick = Probability of receiving the drug and not getting sick + Probability of receiving the placebo and not getting sick
Probability of receiving the placebo:
Out of the total study population, 400 people received the placebo.
Probability of receiving the placebo = Number of people receiving the placebo / Total study population
Probability of receiving the placebo = 400 / (600 + 400)
Probability of receiving the placebo = 0.4 or 40%
Effective rate of the placebo = 10%
Probability of receiving the drug and not getting sick = Probability of receiving the drug x Effective rate of the drug
Probability of receiving the drug and not getting sick = 0.6 x 0.87
Probability of receiving the drug and not getting sick = 0.522 or 52.2%
Probability of not getting sick = 0.522 + 0.4 x 0.1
Probability of not getting sick = 0.522 + 0.04
Probability of not getting sick = 0.562 or 56.2%
To determine the probabilities, we need to use the information given about the effectiveness rates and the number of people in each group.
1) Probability of receiving the drug and not getting sick:
In the group that received the actual medication, the drug was 87% effective. Therefore, out of the 600 people who received the drug, we can expect 87% of them not to get sick. This means 0.87 * 600 = 522 people who received the drug will not get sick.
2) Probability of not getting sick:
To determine the probability of not getting sick, we need to consider both groups: the group that received the drug and the group that received the placebo.
In the group that received the drug:
From the calculation in the previous question, we know that 522 people who received the drug will not get sick.
In the group that received the placebo:
We are given that the effectiveness rate of the placebo, or the group that received the placebo, is 10%. This means that out of the 400 people who received the placebo, we can expect 10% of them not to get sick. Therefore, 0.10 * 400 = 40 people who received the placebo will not get sick.
Now, we can determine the total number of people who will not get sick:
Total number of people who will not get sick = Number of people receiving the drug who will not get sick + Number of people receiving the placebo who will not get sick
= 522 + 40
= 562
To calculate the probability of not getting sick, we divide the total number of people who will not get sick by the total number of people in the study:
Probability of not getting sick = Number of people who will not get sick / Total number of people in the study
= 562 / (600 + 400)
= 562 / 1000
= 0.562
Therefore, the probability that any person in the study will not get sick is 0.562, or 56.2%.