A drug company claims that a new headache drug will bring instant relief in 90% of all cases. If a person is treated with a placebo, there is a 20% chance that the person will feel instant relief. In a clinical trial, half the subjects are treated with the new drug and the other half receive the placebo. If an individual from this trial is chosen at random, what is the probability that the person will have experienced instant relief?
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To find the probability that a randomly chosen individual from the clinical trial will have experienced instant relief, we need to consider two scenarios: when the person received the new drug and when the person received the placebo.
Let's break down the given information:
- The drug company claims that the new headache drug will bring instant relief in 90% of all cases.
- If a person is treated with a placebo, there is a 20% chance that the person will feel instant relief.
- Half of the subjects in the clinical trial were treated with the new drug, and the other half received the placebo.
Now let's calculate the probability for each scenario and then combine them:
1. Probability of experiencing instant relief given the new drug:
According to the drug company's claim, the new drug will bring instant relief in 90% of cases. So, the probability of experiencing relief with the new drug is 90%.
2. Probability of experiencing instant relief given the placebo:
The probability of experiencing relief with a placebo is given as 20%.
Since half of the subjects received the new drug and the other half received the placebo, the probability of selecting an individual who received the new drug is 0.5 (50%), and the probability of selecting an individual who received the placebo is also 0.5.
Now, let's calculate the total probability of experiencing instant relief:
Total probability = (Probability of relief with the new drug) × (Probability of receiving the new drug) + (Probability of relief with the placebo) × (Probability of receiving the placebo)
Total probability = (0.9 × 0.5) + (0.2 × 0.5)
Total probability = 0.45 + 0.1
Total probability = 0.55
Therefore, the probability that a randomly chosen individual from the trial will have experienced instant relief is 0.55, or 55%.