CIMA BA2 Syllabus D. DECISION MAKING - Risk And Probability - Notes ^{1 / 7}

This is present when future events occur with measurable probability.

Risk can be quantified by applying probabilities to the various possible outcomes, because we have past experience.

This is present when the likelihood of future events is incalculable

Uncertainty is an inability to predict the outcome of an activity due to a lack of information or past experience.

Probability

Is the likelihood of a particular outcome from a given event. 

There are 3 sorts of probability:

1. Exact - These apply to a population, for example, drawing a red bull out of a bag full of different coloured red bulls.

2. Empirical - This is derived from recording historical observations, for example, finding the number of visitors coming to a shop everyday.

3. Subjective - This has to do with judgement, for example, finding diamonds in a particular geological area.

Notation

1. The probability of x occurring can be written like this P[x]

   This is:

   = Number of ways x can occur / Number of total outcomes 

   For example, think of rolling a die, the chance of rolling a 6 is: 1/6 = 0.167 or 16.7% chance

2. ∑P = 1 always

   The sum of all probabilities should be 1 always. 

   For example, think of rolling a die, the chance of rolling a 1,2,3,4,5,6 is: 1/6 = 0.167 or 16.7% chance

   So, 16.7% x 6 = 100%

Terminology (AND) ; (OR)

Can these events happen at the same time?

No - this is known as mutually exclusive, for example, think of your country of both, it is either one or the other. The notation for this is OR

Yes - this is known as non mutually exclusive, for example, you can be born in a country and you can be a boy. The notation for this is OR

The next question is:

Are the outcomes linked to each other?

No - This means that they are independent of each other. The notation for this is AND

Yes - This means that they are dependent on each other, for example with twins, one will be the same as each other. The notation for this is AND

When probabilities are independent or dependent, here we are talking about multiplication, the probability of (x and y).

If they are independent, you do this P(x) x P(y)

If they are dependent, you do this P(x) x P(y/x) (This is the probability of x happening, given the probability of y already happening).

Illustration P(x and y) - Independent

What is the probability of getting 2 heads when we flip a coin?

Solution

These are independent, therefore:

P(h) x P(h) = 0.5 x 0.5 = 0.25 or 25%

Illustration - P(x) x P(y/x) - Dependent

Let us say that we have 30 balls in a bag, 16 are red and 14 are blue. 

What is the chance of pulling out 2 reds, one after each other? 

Solution

These are dependent on each other, because the chance of pulling out a first red is 16/30, and the chance of pulling out the second red is 15/29. 

Therefore, the calculation will look like this:

P (R) x P (R/r) = 16/30 x 15/29 = 0.27 = 27% chance of this happening

Illustration - Independent

There is a CIMA student that is taking 2 exams, 

Exam 1: 70% passing chance
Exam 2: 60% passing chance

What is the probability of not passing both?

Solution

30% x 40% = 12%

Illustration - Dependent

A class has 16 males and 14 females. 

What is the probability of picking the first male and the second male also?

Solution

16/30 x 15/29 = 0.27 = 27% chance of this happening

Additive P (x OR Y)

Here, we have 2 events happening at the same time, they cannot happen at the same time (mutually exclusive) or they can happen at the same time (non mutually exclusive). 

If they are mutually exclusive, we calculate it like this:

P (x) + P (y). 

For example, what is the probability of being born in March or April?

1/12 + 1/12 = 1/6 

If they are not mutually exclusive, but now we calculate it like this:

P (x) + P (y) - (P x and y)

For example, what is the probability of being born in March or on a Sunday?

1/12 + 1/7 - (1/12 x 1/7) = 21.4%

Illustration - Mutually Exclusive

Next year sales forecast:

Increase in sales - 10% chance
No change in sales - 70% chance
Decrease in sales - 20% chance 

What is the chance of an increase or no change in sales?

10% + 70% = 80%

Illustration - Not Mutually Exclusive

In a survey of CIMA students, 

40% are doing BA2 
80% of them are using acowtancy for whichever CIMA paper they are doing

As we see, these are not mutually exclusive, you can be a BA2 student and an Acowtancy student. 

What is the probability of being a BA2 student and being an Acowtancy student?

Solution

P (x) + P (y) - (P x and y)

= 40% + 80% - (40% x 80%) = 88%

Complementary Probabilities

This is when we have lots of events, that can be broken down into 2 outcomes - success and failure. 

P (s) = 1 - P (f) 

This means that the probability of success is equals to 1 minus the probability of failure. 

Let's put some numbers to this. 

If we are tossing a coin 4 times, what is the probability of getting at least 1 head?

Solution

What is the probability of failing each time?

1/2 x 1/2 x 1/2 x 1/2 = 1/16

Therefore, now we know the probability of getting at least 1 head is 15/16

Conditional Probabilities

This is the probability of something happening, given that something else has already happened. 

Let's put some numbers to this, 

In a one week study course:

70% of students used Acowtancy and 1/7 showed no improvement
30% of students used other providers and 1/3 showed improvement

This question does not give us a number of people, so we need to make up a number of people for our table. 

Let us use 1,000 people.

What is the probability of someone who used acowtancy and showed improvement?

600/700

What is the probability of someone who did not improve and used the others?

200/300

Illustration - Conditional Probability

There are 100 students.

60% are male, 40 of these pass
40% are female

30 of the students fail in total.

Let's do our table!

What is the probability of a random female who passes?

Solution

30/100 = 30%