I need explanations for Mathematical Proofs.
1 answer:
Answer:
Proofs employ logic expressed in mathematical symbols, along with natural language which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic.
According to Bleiler-Baxter & Pair [22], for a mathematician, a proof serves to convince or justify that a certain statement is true. But it also helps to increase the understanding of the result and the related concepts. That is why a proof also has the role of explanation.
There are many different ways to go about proving something, we'll discuss 3 methods: direct proof, proof by contradiction, proof by induction. We'll talk about what each of these proofs are, when and how they're used. Before diving in, we'll need to explain some terminology.
Although I will focus on proofs in mathematical education per the topic of the question, first and foremost proofs are so hard because they involve taking a hypothesis and attempting to prove or disprove it by finding a counterexample. There are many such hypotheses that have (had) serious monetary rewards available.
In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference.
All mathematicians in the study considered proofs valuable for students because they offer students new methods, important concepts and exercise in logical reasoning needed in problem solving. The study shows that some mathematicians consider proving and problem solving almost as the same kind of activities.
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Answer:
a) Poisson distribution
b) 99.33% probability that in any one minute at least one purchase is made
c) 0.05% probability that seven people make a purchase in the next four minutes
Step-by-step explanation:
In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:
In which
x is the number of sucesses
e = 2.71828 is the Euler number
is the mean in the given time interval.
5 people per minute check out from their shopping cart and make an online purchase.
This means that
a) What model might you suggest to model the number of purchases per minute?
The only information that we have is the mean number of an event(purchases) in a time interval. Each event is also independent fro each other. So you should suggest the Poisson distribution to model the number of purchases per minute.
b) What is the probability that in any one minute at least one purchase is made?
Either no purchases are made, or at least one is. The sum of the probabilities of these events is 1. So
We want to find
So
In which
1 - 0.0067 = 0.9933.
99.33% probability that in any one minute at least one purchase is made
c) What is the probability that seven people make a purchase in the next four minutes?
The mean is 5 purchases in a minute. So, for 4 minutes
We have to find P(X = 7).
0.05% probability that seven people make a purchase in the next four minutes