### 22 May, 2020: Probability 1

This is the third session of the Student Learning Continuity Program for Years 10-12, an educational series made possible by the Ministry of Education and Human Resources Development (MEHRD).

**Presenters:** Lenny Olea and Martin Ruhasia

**Topic:** Probability

Note – For Year 12 (Form 6), your Mathematics Common Assessment Task (CAT) will be on Probability.

**Objectives: **

By the end of this session students should be able to

- Express probability in a variety of ways; as a percentage, a decimal, a fraction or a number out of a total
- Differentiate between equally likely and non-equally likely events
- Relate likeliness of an event to probability from 0 to 1
- Differentiate between mutually exclusive events and independent events
- Identify complementary events
- Use concepts of probability to determine the probability of events in situations where sample with and without replacement takes place.
- Represent the possible outcomes of an experiment on a tree diagram.

**Introduction: **

The concept of probability is part of our everyday lives. It is a household word. We hear it or speak it every day. Weather forecasts are given probabilistically. Most people know that probability is somehow connected with chances. In statements such as:

“It is likely to rain today”.

“I have a reasonable chance of passing this course”.

“He will probably win the tennis competition”.

“I am almost certain that he will be elected”.

We are referring to situations where there is an element of uncertainty about the outcome of a particular situation. In probability theory, we are concerned with assigning a “**measure of likelihood**” or “**probability of occurrence**” to the outcome of an experiment. Probability is the chance that a particular event will occur in a given set of circumstances, depending on the possible events.

Probability lies between 0 and 1. If the probability of an event is 0, we are certainly sure such event will not occur. If the probability of an event is 1, we are certainly sure such event will occur.

Let us look at some events and assign probabilities to it:

“What is the probability that ice will fall in Honiara next week?”

“What is the probability that it will rain tomorrow?”

“A teacher chooses a student at random from a class of 30 boys. What is the probability that the student chosen is a boy?”

We can estimate probabilities from experimental results by using the rule:

**Sampling with replacement and without replacement**

** **__Sampling with replacement__

Eg 1. From a well-shuffled deck of 52 cards one card is selected at random. Find the probability of selecting:

a). a heart b). a picture card c). a red card

**Sampling without replacement**

** **Eg2. A mixed team of five boys and four girls elects a captain and a vice captain.

a). Illustrate this with a tree diagram.

b). Find the probability of electing a boy captain and a girl vice-captain.

c). Find the probability of electing the same sex captain and vice-captain.

**PROBABILITY OF TWO EVENTS**

** ****MUTUALLY EXCLUSIVE EVENTS **

Mutually Exclusive Events:

- cannot happen at the one time
- have no element in common

Example

Consider when a die is rolled. Let **A** be the event that an even number turns up and **B** be the event that an odd number shows.

Sample Space = {1 , 2 , 3, 4, 5, 6 }

Events, A = { 2 , 4 , 6} and B = { 1 , 3 , 5 }

Events A and B have no element in common. An even and odd number cannot occur at the one time on a single toss of a die.

Therefore the events A and B are mutually exclusive.

NB: If events have no element in common, those events are called mutually exclusive events.

** **

**COMPLEMENTARY EVENTS**

If A is any event in a sample space **S** and if **A’ **is the complement of **A**, then

P (A) + P (A’) = 1

P (A) = 1 – P (A’)

Consider when a die is rolled. Let **A** be the event that an even number turns up. The probability that an odd number shows up is

Sample Space = {1, 2, 3, 4, 5, 6}

Event A = {2, 4, 6}

**INDEPENDENT EVENTS**

Two events are called independent events if the outcome of one event has no effect on the outcome of the other.

Example

A coin is tossed and a die is rolled. Let **A** be the event that a HEAD shows on the coin and **B** be the event that a 4 turns up on the die.

Events A and B are independent because the outcome of a 4 does not depend on whether a head has shown on the die

Thank you every students for listening and we hope you grasp some important basic concepts on this topic (Probability) to take with you when you return to school.

Our next session with you will on normal distribution curve and we’ll also explaining What to do when you carry out your Mathematics statistics minor research Project chapter by chapter.

*For your information:*

Form 3, Form 5 and Form 6 Mathematics, we create an account (page) on Facebook for revisions. You can go to that page and post your Mathematics questions and we’ll give you the solution. We are offering you a free service to you do your revisions. All you need to do is to buy your data and use wisely for this purpose.

Once again thank you every good students out there and all best in your studies.

Our contacts: Lenny Olea – 8628229

Martin Ruhasia – 7716696

Stay safe and have a blessed weekend.