Mastering Mathematics Smartly
by Wee Wen Shih

A unique self-help website that provides comprehensive coverage of mathematics at A-level & beyond, written in a student-friendly style.

Probability concepts

Students are often baffled by this question: What's the difference between independent events and mutually exclusive events?

This website explains it and says more about other concepts encountered in probability. Go check it out.

Here is another nice link, specifically on independent & mutually exclusive events.

There is a close relationship between permutations & combinations and probability, which is often assessed. For example, the probability that person A is in front of a line and person B is at the back of a line is given by "number of arrangements whereby A is in front and B is at the back" divided by "number of arrangements without restriction".

Probability versus P&C

I was going through a question about probability with my class and I gave two approaches to solving the problem.

First, let's consider this question:

A basket contains 25 mangoes. Out of these, 20 are sweet and 5 are sour. 3 of the sweet mangoes have worms and only one sour mango contains worms. Two mangoes are chosen at random from the basket. Find the probability that both mangoes are sweet and at least one has worms.

Approach 1: Via P&C (Permutations & Combinations).

Required probability = \frac {^3\text{C}_2 + ^3\text{C}_1 \times ^{17}\text{C}_1}{^{25}\text{C}_2} = \frac{9}{50}.

In the numerator, two cases are taken into account. Firstly, we may have two sweet mangoes with worms. Secondly, we may have two sweet mangoes in which one has worms and the other without.

In the denominator, we simply pick any two mangoes.

Approach 2: Via Probability.

Required probability = \frac{3}{25} \times \frac{2}{24} + 2 \times \frac{3}{25} \times \frac{17}{24} = \frac{9}{50}.

The two terms correspond exactly to the cases described in Approach 1. However in case 2, we need to consider ordering (i.e. worms, no worms or no worms, worms).

Now, it is interesting for us to ask why there is no ordering in Approach 1 but there is so in Approach 2.

Let's look deeply at the working in Approach 2, we can rewrite it as follows:

\frac{3}{\frac{25 \times 24}{2}} + \frac{3 \times 17}{\frac{25 \times 24}{2}} = \frac{^3\text{C}_2 + ^3\text{C}_1 \times ^{17}\text{C}_1}{^{25}\text{C}_2}.
 
Thus, we see that ordering is somehow implied within the working of Approach 1. This goes to show that we need not be very concerned about ordering when we apply P&C concepts to solve this problem. On the other hand, if one chooses to use probability concepts one has to think about ordering, or else he/she risks losing accuracy in the computation.