Scenario three would be one where you have again four people.

The first two people shock, the third person refuses, and the last one shocks.

So the probabilities are 0.65 for the first two, 35 for

the third and 65 for the last one.

Multiplying these probabilities once again gets us to the same answer.

Lastly, scenario number four we again have our four people.

This time we're going to have the first three people

shocking A B and C and D not, refusing to shock.

The probability associated with shocking is 0.65 for the first three.

And 0.35 for the last person.

Once again we multiply the probabilities since these are independent outcomes, and

we're looking for the joint probability, and the answer once again is 0.0961.

So what's going on here?

What we're saying is that, the possible scenarios could be scenario number 1

or scenario number 2 or scenario number 3 or scenario number 4.

These are disjoint scenarios, disjoint outcomes.

They can't all happen at the same time. Therefore when we say or, we add the

probabilities, and therefore we find that the overall probability that exactly one

person out of four refuses to administer the shock is 0.3844.

We could have actually arrived at this answer as the probability

of the first scenario or any scenario times the number of scenarios.

So if we didn't have to go through the scenarios one by one

for illustrative purposes, after we were done

with the first calculation we could quickly

try to figure out how many scenarios there are and simply multiply the probability

of one scenario with the number of scenarios to arrive at the same answer.

This is a perfect setting for the binomial distribution,

as this distribution describes the probability of having exactly

k successes in n independent Bernouilli trials with probability of success, p.

We show that this probability can be calculated as the product

of the number of scenarios times the probability of a single scenario.

The probability of a single scenario is simply p to

the k times 1 minus p to the n minus k.

Let's decipher what this means. This means the probability of success to

the power of number of successes, that was our k.

Multiplied by the probability of failure, to the power of number of failures.

To find the number of scenarios

we actually enumerated each possible scenario,

but this was only feasible since there were only four of them.

And to be frank it was little tedious and boring as well.

If there were many more, say we were looking for how many scenarios for

four success in 100 trials, this method would be very tedious, and

also very error prone. Therefore we usually use an alternative

approach, namely the choose function which is useful for calculating the number

of ways to choose k successes in n trials. To evaluate this function,

we divide n factorial by, by k factorial times

n minus k factorial. Let's give a couple examples here.