## Introduction to probability (A level Statistics)

The probability of an event is the likelihood or chance of that event happening.

The range of values of the probability of any event lie between 0 and 1, where

• 0 is the probability of an event absolutely impossible.
• 1 is the probability of an event absolutely possible.

For example:

• the probability of finding a human being 2cm tall is 0 because it will never happen.
• the probability that the sun will rise tomorrow is 1 because it will always happen.

The probability of an event happening can never be greater than 1 or less than 0, it usually lies somewhere between 1 and 0, and is expressed either as a proper or decimal fraction.

The probability of an event happening is calculated by the following formula:

Probability = $latex \frac{Expected\ Outcome}{Total\ Outcome}$

### Example 1

A fair coin is tossed, find the probability that it will land with the head showing:

P(Head) = $latex \frac{Expected\ Outcome}{Total\ Outcome}$

= $latex \frac{Number\ of\ heads}{Total\ number\ of\ sides}$

= $latex \frac{1}{2}$

### Example 2

A fair 6 sided die is tossed, find the probability that it will land with the face labeled 2:

P(Head) = $latex \frac{Expected\ Outcome}{Total\ Outcome}$

= $latex \frac{Number\ of\ sides\ labeled\ 2}{Total\ number\ of\ sides}$

= $latex \frac{1}{6}$

If take p as the probability of an event happening and q as the probability of the same event not happening, then the total probability is p + q, which is equal to an absolute certainty. Hence p + q = 1.

### Expectation

The expectation, E, on an event is the average occurrence of that event.

From the formula Probability = $latex \frac{Expection}{Total\ Outcome}$

• p = $latex \frac{E}{n}$
• E = pn

we can define expectation of an event happening as the product of the probability, p of an event happening and the number of attempts made, n.

### Example 2

For a fair 6 sided die that is tossed, the probability that it will show an even number score is 0.5. Find its expectation, if the die is thrown 10 times, and explain what the value means.

E = pn

E = 0.5(10)

E = 5

This means that if we throw a die 10 times, we expect to get an even score 5 times.

### Dependent event

A dependent event is an event whose probability of happening affects the probability of another event happening.

Let us say we have a group of 100 people and from which the probability of a random person in the group being male is p1. We later remove 10 people from the group leaving 90. The probability of a random person in the group being male in the group is now p2, which is affected by the genders of the 10 people we removed from group. The probability p2 is dependent on probability p1.

### Example 3

We have a class of 10 mathematics students of which 6 are boys and 4 are girls. A class president is to be chosen from the group. Find the probability that the class president is a:

1. boy
2. girl

After a president is chosen, a vice president is then chosen at random from the remaining students.

Show that the probability of the vice president being a boy or girl depends on the probability of the president being a boy or a girl.

### Solution 3

Probability of the president being a boy:

1. P(Boy) = $latex \frac{E}{n}$
• P(Boy) = $latex \frac{6}{10}$

• P(Boy) = $latex \frac{3}{5}$

2. Probability of the president being a girl:

3. P(Girl) = $latex \frac{E}{n}$
• P(Girl) = $latex \frac{4}{10}$

• P(Girl) = $latex \frac{2}{5}$

Since we have 10 students in the class, if a president is chosen, we are left with a pool of 9 students to chose from.

If the president is a boy, we are are left with 5 boys and 4 girls to choose from. This means that the probability of the vice president being a boy is:

• P(Boy) = $latex \frac{E}{n}$
• P(Boy) = $latex \frac{5}{9}$

and the probability of the vice president being a girl is:

• P(Girl) = $latex \frac{E}{n}$
• P(Girl) = $latex \frac{4}{9}$

If the president is a girl, we are are left with 6 boys and 3 girls to choose from. This means that the probability of the vice president being a boy is:

• P(Boy) = $latex \frac{E}{n}$
• P(Boy) = $latex \frac{6}{9}$

and the probability of the vice president being a girl is:

• P(Girl) = $latex \frac{E}{n}$
• P(Girl) = $latex \frac{3}{9}$

This shows that the probability of the vice president being boy or a girl is dependent on whether the president is a boy or a girl.

### Independent event

An independent event is an event whose probability of happening does not affect the probability of another event happening.

### Example 3

We have a class of 10 mathematics students of which 6 are boys and 4 are girls. A class president is to be chosen from the group. Find the probability that the class president is a:

1. boy
2. girl

After a president is chosen, a quiz captain is then chosen at random from the same 10 students. The president is also allowed to be the quiz captain.

Show that the probability of the quiz captain being a boy or girl does not depend on the probability of the president being a boy or a girl.

### Solution 3

Probability of the president being a boy:

1. P(Boy) = $latex \frac{E}{n}$
• P(Boy) = $latex \frac{6}{10}$

• P(Boy) = $latex \frac{3}{5}$

2. Probability of the president being a girl:

3. P(Girl) = $latex \frac{E}{n}$
• P(Girl) = $latex \frac{4}{10}$

• P(Girl) = $latex \frac{2}{5}$

Since we have 10 students in the class, if a president is chosen, we are still left with a pool of 10 students from which to choose the quiz captain. So the probabilities of the quiz captain being either a boy or a girl do not change because the sample does not change after electing the president.

This shows that the probability of the quiz captain being boy or a girl is is independent of whether the president is a boy or a girl.