Probabilities II: Exclusive and independent events

Two events are exclusive if they cannot happen at the same time. If two events A and B are exclusive, then P(A or B) = P(A) + P(B).

For example, in the experiment of throwing a die, the events "getting an even number" and "getting a '1'" are exclusive, because they cannot occur at the same time: the result cannot be an even number (2, 4 or 6) and AT THE SAME TIME, 1.

Here, if
A = {getting an even number}
B = {getting a '1'}

and we want to find the probability of A or B occurring, then

P(A or B) = P(A) + P(B) = 3/6 + 1/6 = 4/6 = 2/3.


Two events are independent if the occurrence of one doesn't depend on the occurrence of the other. If two events A and B are independent, then P(A and B) = P(A) . P(B)

For example, in the experiment of tossing a coin two times, we can get two possible results (head or tail), then n=2.

If
A = {getting a 'head' in the first toss}
B = {getting a 'tail' in the second toss}

then, A and B are independent (because the event of getting a 'tail' in the second toss is unaffected by the event of getting a 'head' in the first one; they don't relate to each other).

Then, the probability of getting a 'head' in the first toss AND a 'tail' in the second toss is P(A and B) = P(A).P(B) = 1/2 . 1/2 = 1/4.


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PRACTICE FROM THE BOOK: Page 354, Exercise 9.

Probabilities I: simple probability

In simple probability, we can define the following elements:

• event: a situation we want to achieve (e.g.: getting an odd number when rolling a die; picking a card of hearts from a deck of cards).

• possible cases: the set of all the likely cases in an experiment (e.g.: the numbers {1,2,3,4,5,6} in the experiment of rolling a die; the set {head, tail} in the experiment of throwing a coin).

• success cases: the set of all the cases where the event is true (e.g.: if the event is getting an odd number when rolling a die, the success cases is the set {1,3,5}; if the event is picking a card of hearts from a deck of cards, the success cases is the set {A of hearts, 2 of hearts, 3 of hearts, 4 of hearts, ... , 12 of hearts}).

In general we say that, if A is an event, then the probability of A happening is the result of dividing the number of success cases by the number of possible cases:

P(A) = s / n

where:

A is the event;

s is the number of success cases;

n is the number of possible cases.

Events are called with the first capital letters: A, B, C, etc.

Properties:

• A probability lies between 0 and 1.
• If an event cannot happen, the probability of it occurring is 0 (for example, getting a 7 when rolling a die of numbers 1 to 6).
• If an event is certain to happen, the probability of it occurring is 1 (for example, getting a total lower than 15 when rolling two dice and adding the results; because in any throw, the sum is at least 2 (1+1) and at most (6+6), and the sum is always less than 15).
• If P(A) = p, the probability of A not occurring is P(not A) = 1-p (for example, if the probability that today will rain is 0.7, then the probability that today will NOT rain is 1-0.7 = 0.3).

Examples:

1) Consider the experiment of rolling a die. What is the probability of getting:

a) a six?

A = {getting a six}
s = 1 (only success case: getting a '6')
n = 6 (6 possible results)

So, P(A) = 1/6

b) a number greater than 3?

B = {getting a number greater than 3}
s = 3 (success cases: {4,5,6})
n = 6 (6 possible results)

So, P(B) = 3/6 = 1/2

c) a three or a five?

C = {getting a 3 or a 5}
s = 2 (success cases: {3,5})
n = 6 (6 possible results)

So, P(B) = 2/6 = 1/3


2) A bag contains 10 red balls, 5 blue balls and 7 green balls. Find the probability of:

a) getting a red ball: 10/22 = 5/11

b) getting a blue or a green ball: 12/22 = 6/11

c) not getting a green ball:

A = {getting a green ball}
P(A) = 7/22

P(not A) = 1 - P(A) = 1 - 7/22 = 15/22

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PRACTICE FROM THE BOOK: Pages 351-353 (Exercise 8)

Transformations using matrices

A transformation can be represented as a matrix. We can perform translations, reflections, rotations and enlargements by multiplying a transformation matrix M by another matrix P. Matrix M is always a 2x2 matrix and represents the transformation used, while matrix P contains, in each column, the vertices of the figure we want to perform the transformation on.

For example,

The transformation is represented by M, the vertices of the figure are A(1,2); B(0,3) and C(2,-1). The image under the transformation has vertices A'(1,-2); B'(0,-3) and C'(2,1).

This way we can conclude that the transformation is a REFLECTION in the x-axis.

Similarly, we can get rotations, translations and enlargements by always multiplying a transformation matrix by the matrix with the vertices of the figure, and we obtain the image of the figure under the transformation. Then, by looking at the transformation in the graph, we can guess the type of transformation and describe it fully (axis, vector, centre, angle of rotation, etc.).

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PRACTICE FROM THE BOOK: Page 311, exercises 1, 2 and 3.

Bearings

A bearing is a 3-digit angle (for example, 132º, 087º, 315, etc.) from 0º to 360º. It is measured clockwise from the North.

In the following example, we can say that the bearing of B from A is 120º because if we begin from A, the angle measured clockwise from the North is 120º. Similarly, the bearing of A from B is 300º, because if we begin from B, the angle measured clockwise from the North is 300º.

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PRACTICE FROM THE BOOK: Page 201, Exercise 5 (3, 4, 7, 8).