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ProbabilityINDEPENDENT EVENTS
Independent Events
Life is full of random events!
You need to get a "feel" for them to be a smart and successful person.
The toss of a coin, throwing dice and lottery draws are all examples of random events.
Dependent Events
Sometimes an event can affect the next event.
Example: taking colored marbles from a bag: as you take each marble there are less marbles left in the bag, so the probabilities change.
We call those Dependent Events, because what happens depends on what happened before
Independent Events
Independent Events are not affected by previous events
This is an important idea!!
A coin does not ‘know’ that it came up heads before
Each toss of a coin is a perfectly isolated thing
Independent Events
Example: You toss a coin and it comes up "Heads" three times ... what is the chance that the next toss will also be a "Head"?
The chance is simply ½ (or 0.5) just like ANY toss of the coin.
What it did in the past will not affect the current toss!
Independent Events
Some people think "it is overdue for a Tail", but really truly the next toss of the coin is totally independent of any previous tosses.
Saying "a Tail is due", or "just one more go, my luck is due" is called The Gambler's Fallacy
Recall
Probability = Number of ways it can happen
Total number of outcomes
Example
What is the probability of getting a head when tossing a coin?
Number of ways it can happen (1 – Head)
Total number of possible outcomes (2 – Head or Tail)
So the probability is ½ or 0.5
Example
What is the probability of getting a ‘5’ or a ‘6’ when rolling a die?
Number of ways it can happen 2 - (5 and 6)
Total number of outcomes 6 – (1, 2, 3, 4, 5, 6)
So the probability is 2/6 or 1/3
Writing Independent Probabilities
As we have already seen, probabilities exist between 0 and 1
As such, we can represent probabilities as decimals or fractions
We can also represent probabilities as percentages
Two or More Independent Events
What happens when two or more independent events are happening?
For example, what is the probability of tossing a coin 3 times and getting three tails as a result?
How can we show this?
Using a Tree Diagram!
A tree diagram is a diagram used to help figure out the probability when there is more than one events happening
Let’s make a tree diagram for a coin toss
Tree Diagram
T
H
T
H
T
H
T
H
T
H
T
H
T
H
Toss 1 Toss 2 Toss 3Possibili
ties
T – T – T
T – T – H
T – H – T
T – H – H
H – T – T
H – T – H
H – H – T
H – H – H
Coin Flip Frequency
Three Tails 1
Three Heads
1
Two Heads, One Tail
3
Two Tails, One Head
3
Total Flips
8
P(Three Tails) = Frequency / TotalP(Three Tails) = 1/8
Example
Your friend invites you to a movie, saying it starts sometime between 4pm and midnight on Saturday or Sunday. What are the chances that it starts on Saturday between 6 and 8?
Let’s make a tree diagram
Saturday
Sunday
4-6
6-8
8-10
10-12
4-6
6-8
8-10
10-12
P(Saturday 6 – 8) = Number of Saturday 6 – 8
Total Possibilities
P(S 6-8) = 18
Example
A coin is tossed three times
Find the probability of getting at least 2 heads
Example: A coin is tossed three times Find the probability of getting at least 2 heads
T
H
T
H
T
H
T
H
T
H
T
H
T
H
Toss 1 Toss 2 Toss 3Possibili
ties
T – T – T
T – T – H
T – H – T
T – H – H
H – T – T
H – T – H
H – H – T
H – H – H
Coin Flip Frequency
Three Tails 1
Three Heads
1
Two Heads, One Tail
3
Two Tails, One Head
3
Total Flips
8
P(At least two H) = Frequency / Total
P(At least two H) = 4/8 = ½
Two or More Independent Events
We can calculate the chances of two or more independent events by multiplying their individual probabilities
What is the Probability of flipping a coin Tails three times?
P(Tails) = ½
P(Tails 3 Times) = ½ x ½ x ½ = 1/8 or 12.5%
Although the individual was ½, the probability for it happening three times in a row significantly decreased
Two or More Independent Events
Thus, we can calculate the probability of multiple independent events by multiplying
P(A and B) = P(A) x P(B)
For the movie example, P(Saturday) = ½
P (Your Time 6-8) = 2/8
P(Saturday and Your Time) = ½ x 2/8 = 1/8
Example
Two cards are drawn from the top of a well-shuffled deck. What is the probability that they are both black aces?
Example
A die is thrown twice. What is the probability that both numbers are prime?
Example
A code consists of a digit from 0-9 followed by a letter. What is the probability that the code is 9Z?
Example
A bag contains 5 red marbles, 4 green marbles and 1 blue marble. A marble is chosen at random from the bag and not replaced; then a second marble is chosen. What is the probability both marbles are green?
Multiplication Law for Independent Events
When finding probabilities for multiple independent events, we use multiplication to find them
For example, when we are looking for the probability of this event AND this event to happen, we multiply their individual probabilities
Independent Events
There is also the possibility that we are looking for the probability of this event OR this event to happen
When this is the case, we add the probabilities
For example, A bag contains 4 red marbles and 3 green marbles. One marble is drawn at random and then put back. A second marble is drawn. What is the probability that:
a) Both marbles are the same colour
b) As least one of the marbles is green
c) The marbles are different colours
Example
A bag contains 4 red marbles and 3 green marbles. One marble is drawn at random and then put back. A second marble is drawn. What is the probability that:
a) Both marbles are the same colour
b) As least one of the marbles is green
c) The marbles are different colours
P(RR or GG)
Example
A bag contains 4 red marbles and 3 green marbles. One marble is drawn at random and then put back. A second marble is drawn. What is the probability that:
a) Both marbles are the same colour
b) As least one of the marbles is green
c) The marbles are different colours
P(RG or GG or GR)
Example
A bag contains 4 red marbles and 3 green marbles. One marble is drawn at random and then put back. A second marble is drawn. What is the probability that:
a) Both marbles are the same colour
b) As least one of the marbles is green
c) The marbles are different colours
Note: We are looking for the probability of P(RG or GR or GG). In this example, there are only four different possible outcomes.
We also know that the probabilities must add to be 1. So we can also find the probability by doing 1 – P(RR)
Using subtraction to find probability can be easier than finding direct probabilities sometimes
Example
A bag contains 4 red marbles and 3 green marbles. One marble is drawn at random and then put back. A second marble is drawn. What is the probability that:
a) Both marbles are the same colour
b) As least one of the marbles is green
c) The marbles are different colours
P(RG or GR)
Example
Sylvia drives through three sets of lights on her way to work. The probability of each set being green is 0.3. What is the probability that all three sets are green?
