*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.*

## Question π€

**[Maximum marks: 12]**

Youβre 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 (theyβre 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]**

## Solution π€

**a.** Letβs define the discrete random variable $X$ 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 $Y$ 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 $n$. We'll continue using the discrete random variable $Y$ as defined above. That said, the probability of three or more shots being effective should be at least 0.99.

Instead of using $P(Y \geq 3)$ we'll use the complementary event:

That said,

By using a GDC one can find $n > 12.542 \ldots$ which means that the briefcase should contain at least 13 shots.