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.
Example
There are 5 blue balls and 7 red balls. What is the probability of drawing a red ball after a blue ball has been drawn?
The probability of drawing a red ball depends on the first probability of drawing the blue ball. => Dependent probability [step 1]
Prob of first blue ball = 5/12 [step 2a : Find first prob]
Prob of 2nd red ball = 7/11 [step 2b : Find 2nd prob with 11 balls left]
Prob of blue ball, then red ball = 5/12 x 7/11 [step 3 : Compute]
= 0.2651
Complementary Event
The event occurs outside “outside” the outcome.
Example
30% of the students in a swimming class is between 7-10 years old. If a student is picked at random, what is the probability that the student is younger than 7 years old or older than 10 years old?
30% = 30/100 = 3/10
P (age not between 7 - 10) = 1 - 3/10
= 7/10
Using (Venn) diagram
“At least one” Event
~ happens at least once in multiple events
Example
A dice is rolled 3 times. What is the probability of getting at least one 4?
At least once => can be all 3 times, 2 times, 1 time
Thus, at least once is computed by computing not happen at all, and then subtract from the total event.
P (not 4) = 5/6
P (not 4) for 5 times = 5/6 x 5/6 x 5/6
P (at least once) = 1 - P(not 4)3
= 1 - (5/6)3
= 0.4212
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