00:01
For this exercise, we are told that heart failure is due to natural consequences, 80 % of the time, and it is due to outside factors 13 % of the time.
00:13
And now we're asked to consider a sample of 30 patients with heart failure, and for parts a, b, and c, we are asked for the probabilities that certain numbers of these patients have conditions caused by outside factors.
00:29
So for any given patient, the probability that their heart failure, is caused by outside factors is 0 .13.
00:41
And so let's define a random variable, x, as the number of patients.
00:51
This is out of the sample of 30, whose heart failure is caused by outside factors.
01:06
Here x is a binomial random variable.
01:13
That's because each patient has two possible outcomes, either it's caused by outside factors or it is not caused by outside factors.
01:22
And the outcomes for the patients can be considered independent because the outcome for one patient has no bearing on the outcomes for any of the other 29 patients.
01:32
And so each patient can be seen as a bernoulli trial.
01:36
The number of successes in a set number of bernoulli trials is a binomial random variable.
01:44
The probability mass function or the binomial random variable is given by this formula.
01:50
And for part a, we're asked for the probability that exactly three of the patients have conditions caused by outside factors.
02:09
This is the probability that x is equal to 3.
02:13
And using the probability mass function, this is equal to 30, choose 3 times 0 .13 to the exponent 3, times 1 minus 0 .13, which is 0 .87 to the exponent 27.
02:31
And this comes out to a probability of approximately .2077.
02:43
And for part b we are asked for the probability that 3 or more of the patients have heart failures caused by outside factors.
02:52
This is the probability that x is at least 3.
02:57
And can be re -expressed as 1 minus the probability that x is at most 2.
03:06
So we were to use the probability mass function, we would use the summation...