00:01
This question, we are told that 1 in 10 people is left -handed in the us, so that's probability of 0 .1.
00:10
And for part a, we are told that individuals are chosen at random from the population.
00:16
So what they've just described is each person can be considered a bernoulli trial.
00:23
Because if they are chosen at random from the population, they are independent from each other.
00:28
And for each person, there's two possible outcomes.
00:31
Either they are left -handed or they are not.
00:33
So that is basically the definition of a bernoulli trial.
00:37
Now for a1 we are asked what is the probability that we need to pick 12 people from the population until we find a left -handed individual.
00:45
The number of trials required, or the number of bernoulli trials required, to find the first success is called a geometric random variable.
00:55
So let's say the random variable x is the number of people that we select until we find the first left -handed.
01:05
So x is a geometric random variable and it has one parameter.
01:27
That's the probability of success.
01:30
Now the probability mass function for a geometric random variable is given by this formula.
01:50
And so for a1 we want to find the probability that x equals 12.
02:01
So this is 0 .9 to the exponent 11 times 0 .1.
02:15
Comes out to approximately 0 .0314.
02:23
And then part 2 asks what is the probability that we need to pick at least 12 until we find a left -handed person.
02:31
So that's the probability that x is at least 12.
02:39
This is equal to 1 minus, the probability that x is at most 11.
02:47
And so if we were to do this by hand, we would have 1 minus and then we would find the probabilities that x equals 0, 1, 2, and so on.
02:56
Which means that we're going to be writing out this product 12 times, which is a bit tedious.
03:04
So let's use software to solve this.
03:06
We will use excel here.
03:09
So we want the probability that x is at most 11.
03:14
So in excel, we select a cell and type equals to start a formula.
03:18
Now the function that we actually use is called negative binomial, which is just a generalization of the geometric random variable, where you can specify the number of successes in so many trials.
03:33
So for the geometric it's always one success in so many trials...