Probability: Counting Principle
Probability plays an important role in our lives in that it gives us a beautiful headache of choices to choose from.
Imagine if we had the numbers: 1, 2 and 3. In how many ways can we arrange them (order not important)? The arrangements are: 123, 132, 213, 231, 312 and 321. Thus, we have 6 arrangements.
Now, what if on a lunch meal at a local restaurant, you are spoilt for a choice of a meal comprising of:
For starch: pap, rice, mabele, or mesh potatoes
Salad: coleslaw, or chakalaka
Meat: chicken, steak or fish
Drink: water, orange juice, coke or ice tea.
Fruit: banana, apple or pear
Such choices give us a beautiful headache and mathematics helps cooks to arrange their food if any arrangement/choice is permissible. This also helps to decide if the size of the choices is manageable or not.
To be able to cater, the cooks or packers would have to make 288 meals.
The fundamental counting principle says if there are n1 outcomes for event A to occur, and there are n2 outcomes for event B to occur, then the total possible number of outcomes for both events is n1
This can be generalised for n events and the total outcomes is:
n1*n2*n3*...nn
Now the 288 for the example above is calculated as:
4*2*3*4*3=288
Now, let's try few examples
1 For a work schedule, workers have to choose between two shifts: day or night. Every worker has to work for a total of five days in a week. How many options are available for the workers?
Solution: For this question, there are five days (events). Every day/event has two outcomes (day or night). The total is therefore :
2*2*2*2*2=32
2 For an interview, Thato wants to wear pants, a shirt and heels. She has four pants to choose from, three shirts and five pairs choose from. How many arrangements are possible?
Solution: There three event, namely: pants, shirts and heels. Each have four, three and five outcomes, respectively. The total arrangements are:
4*3*5=60
3 To create a serial number for a school's fundraising concert, the maths students have to create a serial number with the form, LLDDD, where L is the letter with choices between S and A, and D is for digits 1-9. How many serial numbers are possible if repetition is allowed?
Solution: There are two different events, L and D, each occurring two and three times, respectively. For the L, there are two outcomes and for the D, there are 9 outcomes. Therefore, the total serial numbers that can be made are:
2*2*9*9*9=2 916