What is Probability?
- the likelihood that an event will happen from all the possible outcome.
What is an event?
An “event” can be one or more outcomes.
A dice has 6 faces - 1, 2, 3, 4, 5 and 6.
What a dice is tossed, one of six events can happen:- the number is 1, 2, 3, 4, 5, or 6
Probability
Probability = Positive events / Total number of possible outcomes
Example
When a dice is tossed, what is the probability of having the number 2?
Dice Number 1 2 3 4 5 6
Number(Event) 1 1 1 1 1 1
Probability 1/6 1/6 1/6 1/6 1/6 1/6
=>
Total number of events = 6 [Step 1 - the total possible outcome]
Number 2 happens = 1 [Step 2 - the count of positive event ]
Probability of number is 2 = 1/6 [Step 3 - Compute the probability]
Also, probability is
~ expressed as a value between 0 and 1.
~ for an event that will not happen at all (0%), the probability = 0
~ for an event that definitely will happen (100%), the probability = 1
~ Sum of Probability of all possible events = 1
Example
There are 2 red balls and 1 blue ball in a basket. What is the probability of picking up a red ball from the basket?
Colour R R B
Event 1 1 1 (total = 3)
Probability 1/3 1/3 1/3
Event of Red = 2, Probability = 1/3+ 1/3 = 2/3 or 2/3
=>
Total events = 3 [Step 1 - total : picking a red ball, red ball, blue ball]
Picking a red ball = 2 [Step 2 - the count of positive event = 2]
Probability = 2/3 [Step 3 - Compute the probability]
Probability with a Frequency Distribution Table
Example
The following is the data of people going to Johore Bahru
Transport Frequency
Walk 40
Train 24
Bus 56
What is the probability of a person walking to Johore Bahru?
Number of people walking = 40 [Step 1: Happening/positive events]
Total number of people = 40 + 24 + 56 [Step 2 : Total outcome]
= 120
Probability = 40/120 = 1/3 [Step 3 : Positive event / Total outcome]
Probability with Different type of events
Mutually Exclusive
~ cannot happen at the same time
Using (Venn) diagram to represent:
Tossing a coin, Head and Tail are mutually exclusive because the outcome of having a head and tail cannot happen at the same time.
Example
There are 5 blue balls , 3 red balls and 2 yellow balls. What is the probability that either a blue ball of yellow ball is picked?
[either… or… => both events to be added]
Picking a blue ball or a yellow ball = 5 + 2 [Step 1 : Positive happening]
Total outcome = 5 + 3 + 2 = 10 [Step 2 : Total outcome]
Probability = 7/10 [Step 3 : Calculate ]
==> Probability for mutually exclusive event
P(A or B) = P(A) + P(B)
Alternative solution
Prob of picking blue ball = 5/10
Prob of yellow ball = 2/10
Probability = 2/10 + 5/10 = 7/10
Example (with percent)
[At times, percent is used to represent probability. 17% => 17/100 probability]
In a lucky draw, 10% of the prizes are $5 voucher, 15% are $2 voucher and the rest are $1 voucher.
- What is the probability of getting either a $5 voucher or a $2 voucher?
Probability = 10/100 + 15/100
= 25/100
= 1/4
(2) What is the probability of getting $1 voucher?
$1 voucher (Not getting $5 or $2) = 100% - 10% - 15%
= 75%
Probability = 75/100 = 3/4
Not Mutually Exclusive
~ can happen at the same time
Using (Venn) diagram to represent:
Out of 32 students, 20 students play basketball and 12 students play tennis. There are 4 students who play both tennis and basketball.
[4 students playing both basketball and tennis make the event not mutually exclusive]
Using (Venn) diagram to represent the event
Probability of non-exclusive event:
P(A or B) = P(A) + P(B) - P(A and B)
Playing basketball only = 20 - 4 = 16
Probability of playing basketball only = 16/32 = 12
(2) What is the probability of a student playing either tennis or basketball?
Prob of student playing either tennis or basketball = 20/32 + 12/32 - 4/32
= 28/32
= 7/8
Independent event
Each event is not affected by other events.
Two dices were thrown. The outcome of each dice does not affect the other dice.
When both event A and B are independent, the probability of both occurring is
Probability of independent event : P (A and B) = P(A) x P(B)
Example
A classroom has two fans. The probability that a fan is not working is 0.05.
<Step 1: Type of events? Independent because working of both fans are not related>
Probability of a fan not working = 0.5
Probability of a fan working = 1 - 0.5 = 0.95 [probability of positive / happening]
Probability of both fans working = 0.95 x 0.95
= 0.9025
(2) What is the probability that both fans are not working
<step 1 : type of event? Independent - P(A) x P(B) >
Prob a fan not working = 0.5 [step 2 : prob of happening]
Prob of both fans not working = 0.5 x 0.5 [Step 3 : Calculate]
= 0.25
(3) What is the probability either one is not working?
<Step 1 : Type of events : Independent >
Probability of either one is not working [Step 2 : No of events and probability]
= fan A working x fan B not working + fan A not working x Fan B working
= 0.95 x 0.05 + 0.05 x 0.95
= 0.095
Dependent event (conditional)
~ affected by the other event.
Example
There are five blue balls and one red balls in a basket. If a red ball is taken out of the basket, then the second time a ball is to be drawn there is no chance to throw another red ball.
=> the first event affects the outcome of the second event.
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