Probability

The 68-95-99.7 rule is applicable for a Normal distribution.

The Normal Distribution is determined by two parameters: the mean (m) and the standard deviation (s). They are normally denoted by Greek letters but there is no telling what this rubbish browser will do to them.

Anyway, these are used to calculate z-scores for observations. The z score for an observation x is (x - m)/s.

The rule is that

68% of the observation lie within -1 < Z < 1 : that is within 1 standard deviation of the mean;

95% of the observation lie within -2 < Z < 2 : that is within 1 standard deviation of the mean;

68% of the observation lie within -3 < Z < 3 : that is within 1 standard deviation of the mean.

Also, the Normal distribution is symmetrical so 50% of the observations are for z < 0.

Now, 1 - 68% is outside of -1 < Z < 1. That is, the two regions Z < -1 and Z > 1 = 32% and so, by symmetry, each is half as much which is 16%. Thus the first of the above rule also implies the following probabilities:

• 16% for Z < -1 and so 84% for Z > -1
• 16% for Z > 1 and so 84% for Z < 1

Now, given that m = 70 and s = 15

(a) X < 55 => Z < (55 - 70)/15 = -1

So prob (X < 55) = 0.16

(b) X < 40 => Z < (40 - 70)/15 = -2

and so, prob(X < 40) = 2.5%

(c) X < 85 => Z < 1

S0 prob (X < 85) = 84%

(d) Prob(X < 100) = 97.5%

(e) Prob(X > 85) = 16%

Circle Theorems

A line joining the centre of a circle to any of the points on the circle is known as a radius.

The circumference of a circle is the length of the circle. The circumference of a circle = 2 × π × the radius.

The red line in the second diagram is called a chord. It divides the circle into a major segment and a minor segment.

Angles formed from two points on the circumference are equal to other angles, in the same arc, formed from those two points.

Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle. So c is a right angle.

We can split the triangle in two by drawing a line from the centre of the circle to the point on the circumference our triangle touches.

We know that each of the lines which is a radius of the circle (the green lines) are the same length. Therefore each of the two triangles is isosceles and has a pair of equal angles.

But all of these angles together must add up to 180°, since they are the angles of the original big triangle.

Therefore x + y + x + y = 180, in other words 2(x + y) = 180.

and so x + y = 90. But x + y is the size of the angle we wanted to find.

A tangent to a circle forms a right angle with the circle's radius, at the point of contact of the tangent.

Also, if two tangents are drawn on a circle and they cross, the lengths of the two tangents (from the point where they touch the circle to the point where they cross) will be the same.

The angle formed at the centre of the circle by lines originating from two points on the circle's circumference is double the angle formed on the circumference of the circle by lines originating from the same points. i.e. a = 2b.

You might have to be able to prove this fact:

OA = OX since both of these are equal to the radius of the circle. The triangle AOX is therefore isosceles and so ∠OXA = a

Similarly, ∠OXB = b

Since the angles in a triangle add up to 180, we know that ∠XOA = 180 - 2a

Similarly, ∠BOX = 180 - 2b

Since the angles around a point add up to 360, we have that ∠AOB = 360 - ∠XOA - ∠BOX

= 360 - (180 - 2a) - (180 - 2b)

= 2a + 2b = 2(a + b) = 2 ∠AXB

This diagram shows the alternate segment theorem. In short, the red angles are equal to each other and the green angles are equal to each other.

You may have to be able to prove the alternate segment theorem:

We use facts about related angles:

A tangent makes an angle of 90 degrees with the radius of a circle, so we know that ∠OAC + x = 90.

The angle in a semi-circle is 90, so ∠BCA = 90.

The angles in a triangle add up to 180, so ∠BCA + ∠OAC + y = 180

Therefore 90 + ∠OAC + y = 180 and so ∠OAC + y = 90

But OAC + x = 90, so ∠OAC + x = ∠OAC + y

Hence x = y

If the radius of the circle is r,

Area of sector = πr2 × A/360

Arc length = 2πr × A/360

In other words, area of sector = area of circle × A/360

arc length = circumference of circle × A/360

Each roll of the coin can either have a head or not have a head. So in three rolls there could be:

• no heads (all tails);
• 1 head (either first, second or third roll was the head, the other two being tails);
• 2 heads (either first, second or third roll was the tail, the other two being heads); or
• 3 heads (all heads)

giving a sample space for the number of heads in three rolls of a coin:

{0, 1, 2, 3}

Here are some:

• If you have a bag of 10 marbles, 6 of which are red, the chance of taking out a red one at random is 3/5
• If you roll a dice, the chance of getting an even number is 1/2
• If you take the top card of a shuffled deck the chance of it being a diamond is 1/4. The chance of it being a king is 1/13

Each roll has (6 x 6) = 36 possible outcomes.

A 7 or 11 can be rolled in 8 ways:

1--6, 6--1

2--5, 5--2

3--4, 4--3

5--6, 6--5

The probability is (8/36) = 2/9

Doubles can be rolled in 12 ways:

1--1 and reverse

2--2 and reverse

3--3 and reverse

4--4 and reverse

5--5 and reverse

6--6 and reverse

The probability is (12/36) = 1/3.

The probability of both of these rolls happening consecutively is (2/9 x 1/3) = 2/27 = 7.41% (rounded)