Probability

We use probability to describe uncertain events. When you accidentally drop a slice of bread, you don’t know if it’s going to fall with the buttered side facing upwards or downwards. When your favourite sports team plays a game, you don’t know whether they will win or not. When the weatherman says that there is a 40% chance of rain tomorrow, you may or may not end up getting wet. Uncertainty presents itself to some degree in every event that occurs around us and in every decision that we make.

We will see in this chapter that all of these uncertainties can be described using the rules of probability theory and that we can make definite conclusions about uncertain processes.

Probability or possibility or chance is quantitative measure of the chance of occurrence of a particular event

Experiment

An operation which can produce some well-defined outcomes is called an experiment.

Random Experiment:

An experiment in which all possible outcomes are know and the exact output cannot be predicted in advance, is called a random experiment.

Examples:

i. Rolling an unbiased dice.

ii. Tossing a fair coin.

iii. Drawing a card from a pack of well-shuffled cards.

iv. Picking up a ball of certain color from a bag containing balls of different colors.

When we throw a coin, then either a Head (H) or a Tail (T) appears.

A dice is a solid cube, having 6 faces, marked 1, 2, 3, 4, 5, 6 respectively. When we throw a die, the outcome is the  number that appears on its upper face.

A pack of cards has 52 cards.

Outcome

An outcome of an experiment is a single result of that experiment.

Experiment 1: A coin is tossed and it lands with either heads (H) or tails (T) facing upwards. An example outcome of tossing a coin is that it lands with heads facing up:

Experiment 2: Two dice are rolled and the total number of dots added up. An example outcome of rolling two dice:

Experiment 3: Two teams play a soccer match and we are interested in the final score. An example outcome of a soccer match:

Sample space

The sample space of an experiment is the set of all possible outcomes of that experiment. The sample space is denoted with the symbol S  and the size of the sample space (the total number of possible outcomes) is denoted with n(S). Even though we are usually interested in the outcome of an experiment, we also need to know what the other outcomes could have been. Let’s have a look at the sample spaces of each of our three experiments.

Experiment 1: Since a coin can land in one of only two ways (we will ignore the possibility that the coin lands on its edge), the sample space is the set S={H;T}. The size of the sample space of the coin toss is n(S)=2

Experiment 2: Each of the dice can land on a number from 1 to 6. In this experiment the sample space of all possible outcomes is every possible combination of the 6 numbers on the first die with the 6 numbers on the second die. This gives a total of n(S)=6×6=36 possible outcomes. The figure below shows all of the outcomes in the sample space of rolling two dice:

Experiment 3: Each soccer team can get an integer score from 0 upwards. Usually we don’t expect a score to go much higher than 5  goals, but there is no reason why this cannot happen. So the sample space of this experiment consists of all possible combinations of two non-negative integers. The figure below shows all of the possibilities. Since we do not limit the score of a team, this sample space is infinitely large:

Question

A learner wants to understand the term “event”. So the learner rolls 2 dice hoping to get a total of 8. Which of the following is the most appropriate example of the term “event”?

• event set ={(4; 4)}
• event set ={(2; 6); (3; 5); (4; 4); (5; 3); (6; 2)}
• event set ={(2; 6); (6; 2)}

Solution

We recall the definition of the term “event”:An event is a specific set of outcomes of an experiment that you are interested in.Therefore the most appropriate example of the term “event” is event set ={(2;6);(3;5);(4;4);(5;3);(6;2)}.

Question

A learner wants to understand the term “sample space”. So the learner rolls a die. Which of the following is the most appropriate example of the term “sample space”?

• {1; 2; 3; 4; 5; 6}
• {H; T}
• {1; 3; 5}

Solution

We recall the definition of the term “sample space”: The sample space of an experiment is the set of all possible outcomes of all experiment.

Therefore the most appropriate example of the term “sample space” is {1;2;3;4;5;6}

Question

A learner finds a textbook that has 100 pages. He then selects one page from the textbook. What is the probability that the page has an odd page number? Write your answer as a decimal (correct to 2 decimal places).

Solution:

Question