Of 100 patients with a certain disease, 10 were chosen at...

Of 100 patients with a certain disease, 10 were chosen at random to undergo a drug treatment that increases the cure rate from $50 \%$ for those not given the treatment to $75 \%$ for those given the drug treatment. a. What is the probability that a randomly chosen patient both was cured and was given the drug treatment? b. What is the probability that a patient who was cured had been given the drug treatment? c. What is the probability that a specific group of 10 patients was chosen to undergo the drug treatment? (Leave your answer in terms of factorials.)

Of 100 patients with a certain disease, 10 were chosen at random to undergo a drug treatment that increases the cure rate from $50 \%$ for those not given the treatment to $75 \%$ for those given the drug treatment.
a. What is the probability that a randomly chosen patient both was cured and was given the drug treatment?
b. What is the probability that a patient who was cured had been given the drug treatment?
c. What is the probability that a specific group of 10 patients was chosen to undergo the drug treatment? (Leave your answer in terms of factorials.)

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Key Concepts

Joint Probability

Joint probability is the likelihood of two events occurring simultaneously. In problems like this, it involves finding the intersection of two events (for example, a patient both receiving the treatment and being cured) by multiplying the probability of one event by the conditional probability of the second event given the first.

Conditional Probability

Conditional probability is the chance of an event occurring given that another event has already occurred. For instance, determining the probability that a patient who was cured had actually been given the drug treatment involves applying the definition of conditional probability, which compares the joint probability of being cured and receiving treatment with the total probability of being cured.

Combinatorial Analysis

Combinatorial analysis involves counting the number of ways events can occur using combinations or factorial functions. In problems where a specific subset is chosen from a larger group, such as selecting 10 patients out of 100, combinatorial methods are used to calculate probabilities based on the total number of possible selections.

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