Binomial distributions are used for independent experiments (when the probability of success does not change with a previous outcome) and when there are only two possible outcomes and a finite number of trials. Binomial distributions are a subset of discrete random variables, one of the most important topics in the IB HL Math curriculum. As a consequence, it shows up very often on IB Exams.
Some binomial distributions' exercises will involve normal distributions (continuous random variables) as well, however that is not the case for this exercise. Not to say that this is a trivial one.
[Maximum marks: 12]
You’re escaping a zombie attack that was initiated due to the spread of a virus that causes zombisease (one which turns people into zombies). In order to escape, you enter a dark room. Suddenly you notice that, in that room, there are 9 other people. You are stuck inside that room with them. You see a light switch. You turn it on.
a. Given that, in this apocalyptic scenario, there is a 19% chance of someone having become a zombie, what is the probability that, from the other 9 people in the room, at most three of them are zombies. [3 marks]
b. Now, assume three of those people are zombies. You found a suitcase with 5 vaccines against zombisease. The label reads that its efficacy is 52%. Assume that:
- you can tell which people are zombies and which ones are not (zombies look like zombies)
- you can give zombies a shot or more than one shot (they’re friendly in that sense)
i. What is the probability that you are able to cure every zombie with the available shots? [4 marks]
ii. How many shots would there need to be in the briefcase in order for the probability of everyone getting cured is at least 99%? [5 marks]
a. Let’s define the discrete random variable as being the number of people that became zombies (were infected and developed zombisease). The exercise asks for the probability of at most three of them are infected, therefore we should include three in it.
Using the TI-84 one should use the following calculator command
2nd > DISTR > B: binomcdf( binomcdf(9,0.19,3)
b.i. From all the shots available on the briefcase, at least three of them should work. If we define the discrete random variable as the number of shots that are effective, we can find the probability that at least three of them will work as
The TI-84 command is
2nd > DISTR > B: binomcdf( 1 - binomcdf(5,0.52,2)
b.ii. Let's define the number of available shots as . We'll continue using the discrete random variable as defined above. That said, the probability of three or more shots being effective should be at least 0.99.
Instead of using we'll use the complementary event:
By using a GDC one can find which means that the briefcase should contain at least 13 shots.