The number of times that a person contracts a cold in a...

The number of times that a person contracts a cold in a given year is a Poisson random variable with parameter $\lambda=5 .$ Suppose that a new wonder drug (based on large quantities of vitamin $\mathrm{C}$ ) has just been marketed that reduces the Poisson parameter to $\lambda=3$ for 75 percent of the population. For the other 25 percent of the population, the drug has no appreciable effect on colds. If an individual tries the drug for a year and has 2 colds in that time, how likely is it that the drug is beneficial for him or her?

Transcript

00:02 So here we are told that the number of codes contracted by a person in a year is a plus on random variable with parameter lambda equals 5.

00:13 We have also been told that there is now a medicine available, and the probability that the medicine is beneficial, call that b, is 0 .75.

00:28 And the probability that it is not beneficial, call that b prime, is therefore 0 .25.

00:37 Now, if it is beneficial, x is actually a plus on random variable with parameter lambda equals three.

00:51 Otherwise, it remains a plus on with lambda equals five.

01:01 And so we're told that an individual tries the drug for a year and has two codes in that year.

01:11 So what is the probability that the drug was beneficial for this person? so stating that mathematically, it's what is the probability that we had a beneficial drug, that this person derived benefits from the drug, given that the number of codes was two.

01:33 Now, we can use bayes theorem to solve this problem because just looking at the question is somewhat more obvious that we can find the probability that x is two, given that the drug is beneficial.

02:08 So this is thanks to bayes theorem.

02:18 And the reason why i say that it seems, from the question that it's easier to solve this, than this probability is because we know the distribution for the number of codes given that the drug is beneficial.

02:46 So let's start off by solving this factor.

02:55 Probability that we get two codes given that the drug is beneficial.

03:06 This is a plus on random variable with parameter lambda equals 3, and so this is equal to e to the minus 3.

03:20 Times 3 squared over 2 factorial.

03:29 And i'm getting that from the probability mass function for a plus on random variable.

03:52 And this comes out to 0 .2240.

04:05 Now for the probability of the drug being beneficial, we have that in the question that's 0 .75...