Recall the binomial theorem.
1. The binomial expansion of is
Observe that
When we multiply these by ,
• and
combine to make
• and
combine to make
and the sum of these terms is
2. The binomial expansion is
We get the term when
:
Since the sum of the probabilities of all possible outcomes must be 100%, we can deduce the following:
In fact, the probabilities of cooking in less than 20 or more than 30 sum up to 15%, which means that the remaining outcome (i.e. cooking time between 20 and 30) must complete this probability to 15, and in fact 15+85=100.
That being said, all three answers are simply a combination of these three scenarios: let C be the cooking time, for aesthetic reasons:
