BUSM - Lecture 06 - Probability

7/28/2019 BUSM - Lecture 06 - Probability

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NCC Education LimitedV1.0

Business Mathematics

Lecture 6:

Probability


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Probability Lecture 6 - 6.2


Scope and Coverage

This topic will cover:

Calculating probabilities

Relative frequency

Mutually exclusive events

Independent events

Conditional probability


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Learning Outcomes

By the end of this topic, students will be able to:

Use relative frequency to estimate probabilities

Understand the meaning of mutually exclusive

outcomes and be able to calculate probabilities Understand the meaning of independent events and be

able to calculate probabilities of them happening

To use the OR and AND rule

Use tree diagrams to calculate probabilities

Calculate conditional probabilities


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Is This Fair?

Coin Game

I flip two coins. If both land heads, you win.

If they are different, I win.If they are both tails, we flip again!

Is this fair? Discuss.


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Probability Probabilities are written as fractions or decimals, and less

often as percentages

An event can have several possible outcomes

Each outcome has a probability or chance of occurring

When a fair dice is thrown there is equal chance of throwingeach number. The outcomes from the event throwing a

dice are equally likely outcomes

If the outcomes of an event are equally likely, the probability

can be calculated using:

Probability of an event =Number of successful outcomes

Total number of possible outcomes


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Relative Frequency For some events, probability cannot be calculated using equally

likely outcomes

For example, the probability of a train from Newcastle to

Manchester being late. Being late and being on time may not

be equally likely

The probability can be estimated using results of an experimentor a survey by finding the relative frequency

Number of times the event occursin an experiment (or survey)

Total number of trials in the experiment

(or observations in the survey)

Relative Frequency =

Probability based on relative frequency is called experimental probability.

Probability calculated from equally likely outcomes is called theoreticalprobability.


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Examples

1. Bobby drops a drawing pin and records whether itlands point up or down. She repeats the trial 100

times.

The number of times the drawing pin lands point up is 28.

a) What is the relative frequency of the drawing pinlanding point up?

b) Estimate the probability that the drawing pin lands

point down

2. An ordinary dice is thrown 600 times. The dice lands

on a square number 120 times. Is the dice fair?


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Mutually Exclusive Events

Outcomes that cannot happen at the same time are calledmutually exclusive outcomes

E.g. A dice is rolled. It shows 5. It is rolled again. It

shows 2. These events cannot happen at the same time.

They are mutually exclusive

The total probability of mutually exclusive outcomes is 1.

An event cannot happen and not happen at the same time

The sum of the probabilities of mutually exclusive

outcomes is 1 Probability of rolling a 5

= 1 Probability of NOT rolling a 5.


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The OR Rule

If two events, A and B are mutually exclusive:

P(A or B) = P(A) + P(B)

This is known as the OR rule or addition rule for mutually

exclusive probabilities.

Example:

A bag contains 1 yellow, 3 green, 4 blue and 2 red marbles.

a) What is the probability of picking a green or a blue marble?

b) What is the probability of not picking a red marble?


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Independent Events

Two events are independent when the probability of oneevent happening is not affected by the outcome of the other

event

E.g. Roll a dice and flip a coin.

Event A: the dice shows an odd numberEvent B: the coin shows tails.

These events are independent. Neither outcome can

influence the other.

Are these events independent?

You go outside. Event A: It is snowing Event B: It is cold


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The AND Rule

To find the probability of two independent eventsboth happening, multiply the individual probabilities

together

If A and B are individual events

P(A and B) = P(A) x P(B)

This is the AND rule or multiplication rule.

E.g. The probability of rolling an odd number on a

dice AND flipping a coin to get tails is:P(Odd) x P(Tails) = x =


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Example

1. The probability that Steve walks to work on a given

day is 2/3.

a) What is the probability he walks to work on two

consecutive days?b) What is the probability on 2 consecutive days, Steve

walks to work on one and doesnt walk on the other?

Assume consecutive days are independent.


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Tree Diagrams

Independent events and their probabilities can be

shown on a tree diagram. Each event is

represented by a branch

E.g. A coin is flipped twice. Draw a tree diagram to

show all the possible outcomes


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A coin is flipped twice. Draw a tree diagram to show allthe possible outcomes. Outcomes Probabilities

HH x =

HT x = TH x =

TT x =

1st flip 2nd flip

H

H

T

T

TH


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A box contains 3 red ties and 2 white ties. John picks a tie andputs it on in the morning and puts it back at night. Draw a treediagram to show the possible outcomes over two days.

Outcomes Probabilities1st tie 2nd tie

R

R

W

W

2/5

2/5

W

R

2/53/5

3/5

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1) What is the probability he wears a red tie 2 days running?

2) What is the probability he wears a white 2 on at least one ofthe next 2 days?


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A box contains 3 red ties and 2 white ties. John picks a tie andputs it on in the morning and puts it back at night. Draw a treediagram to show the possible outcomes over two days.

Outcomes Probabilities

RR 3/5 x 3/5 = 9/25

RW 3/5 x 2/5 = 6/25

WR 2/5 x 3/5 = 6/25

WW 2/5 x 2/5 = 4/25

1st tie 2nd tie

R

R

W

W

2/5

2/5

W

R

2/53/5

3/5

3/5

Answers:

1) P(RR) = 9/25

2) P(at least 1 white) = P(RW + WR + WW) = 9/25 + 6/25 + 6/25

= 21/25


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Conditional Probability

When two events are not independent, theprobability of one happening is affected by the

other. This is conditional probability

E.g. A fair dice is rolled. If you know the outcome

is even, what is the probability it is: a) a 4 b) a 5? Sampling without replacement:

If there are a certain number of coloured counters

in a bag and one is picked, but not replaced, theprobability of the next counter to be picked will be

affected by the first counter picked.

P b bili L 6 6 18


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Example

George keeps his clean socks in a bag. There are ten blackand six white socks in the bag.

He takes two socks from the bag one after the other at randomand puts them on without looking.

a) Draw a tree diagram to show the possible outcomes for thecolours of each sock and their probabilities.

b) Use the tree diagram to find the probability that he iswearing:

i) a matching pair ii) one black and one white sock

iii) at least one white sock

P b bilit L t 6 6 19


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Outcomes Probabilities

BB 10/16 x 9/15

= 90/240 = 3/8

BW 10/16 x 6/15= 60/240 = 1/4

WB 6/16 x 10/15

= 60/240 = 1/4

WW 6/16 x 5/15

= 30/240 = 1/8

1st sock 2nd sock

B

B

W

W

6/16

5/15

W

B

6/1510/16

9/15

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For the probabilities for the second sock, think about how many

socks are left after the 1st sock has been taken.

a) P(matching)= P(BB) + P(WW)

= 3/8 + 1/8 = 4/8 = 1/2

b) P(one black, one white)

= P(BW) + P(WB) = 1/4 + 1/4 =1/2

c) P(at least one white)

= P(BW) + P(WB) + P(WW)

= 1/4 + 1/4 + 1/8 = 5/8

P b bilit L t 6 6 20


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Recap

Explain the following terms: Equally likely outcomes

Relative frequency

Mutually exclusive outcomes Independent events

OR and AND rule

Tree diagram

Conditional probability

Probability Lecture 6 6 21


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Lecture 6 Probability

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BUSM - Lecture 06 - Probability