Transcript Three Events

How to Calculate and Understand
Two and Three Independent
Events
Clayton Flanders Grade 10
BB&N AP Statistics 2004
• Here is a quick overview of how to
calculate the probability of two independent
events.
• However, first some definitions.
• A random experiment is any procedure or
situation that produces a definite outcome
that may not be predictable in advance.
• Example - Seeing how many heads you get
flipping a coin 4 times.
• An outcome is a single possible result of a
random experiment.
• Example - the 2nd flip of the coin landing
heads up.
• An event is any collection of outcomes.
• Ex. - The last 3 flips of the coin had two
heads up and one tails up.
• Two events are said to be independent if the
outcome of one event does not help us
determine how likely the other event is.
•
Definitions taken from the book Statistics and Data Analysis by Andrew F. Siegel and
Charles J. Morgan
One event OR another
• OR is a connector which combines the
outcomes of two events.
• Example- The outcome of wearing a red hat
OR a green shirt is fulfilled when wearing a
red hat by itself, a green shirt by itself, and
it is also fulfilled by wearing both a red hat
and a green shirt at the same time.
One event AND another
• AND is another connector which is fulfilled
when both events happen simultaneously.
• Example- wearing a red hat AND a green
shirt can happen only if a red hat is worn
while a green shirt is worn at the same time.
How to calculate an AND
statement
An AND statement such as P(A and B) is calculated by multiplying the
P(A) by the P(B). The resulting probability is the probability of these two
events happening at the same time.
Example- The probability of Erik Johnson wearing a
green shirt (GS) is .34. The probability of him wearing a
red hat (RH) is .12. What is the probability of Erik
Johnson wearing both a green shirt and a red hat?
P(GS and RH)=P(GS) x P(RH)
P(GS and RH)= .34 x .12
P(GS and RH)= .0408
Note - all these example problems assume that the two
events are independent from one another and not
mutually exclusive
The relationship between AND
and OR
P(A)
P(B)
P(A) + P(B)
1 2 1
Note- the numbers above correspond to how many times that
particular area of the figure is counted in the overall
probability, which means unless all the numbers in the figure
are 1,then it is not an OR statement. Since the P(A and B) is
counted twice in the figure on the right, the P(A and B) must
be subtracted once to make it the P(A or B).
The relationship between AND
and OR
P(A) + P(B)
1 2 1
P(A and B)
P(A or B)
1 1 1
Now that none of the spaces in the figure overlap, it is the
P(A or B), and no longer the P(A) + P(B). This leads us to
the formula...
The relationship between AND
and OR
P(A or B)
P(A)
P(B)
P(A and B)
Example: The probability of Erik Johnson wearing a
green shirt (GS) is .34. The probability of him wearing
a red hat (RH) is .12. What is the probability of Erik
Johnson wearing either a green shirt or a red hat?
P(GS)=.34
P(RH)=.12
P(GS and RH)
= .0408
P(GS or RH)
= .4192
2nd way to calculate an OR
statement
• Now lets come up with another way to calculate
an OR statement.
• First, lets say that event A has a chance to occur,
and event A occurs.
• Since the requirement for OR is fulfilled if either
event A or event B occurs, since event A occurs,
then event (A or B) has happened.
2nd way to calculate an OR
statement
• Now think of what happens if event A doesn’t
occur .
• The requirements for OR is not fulfilled yet.
(which means event (A or B) hasn’t happened yet)
• Now event B has a chance to occur to make the
event (A or B) happen.
• So a 2nd way to calculate OR is P(A or B)=
P(A) + (P(A not occurring) x P(B))
• ( note P(A not occurring)=1-P(A) )
Now calculate the P(A or B) in the same example with the new
formula.
Example Problem: The probability of Erik Johnson wearing a
green shirt (GS) is .34. The probability of him wearing a red hat
(RH) is .12. What is the probability of Erik Johnson wearing
either a green shirt or a red hat?
P(GS or RH) = P(GS) + (P(not a GS) x P(RH))
P(GS or RH)= .34 + ((1-.34)x(.12))
P(GS or RH)=.4092
• We have now covered how to calculate the
probability of two events in a AND or OR
situation.
• We will now move on to three events.
• Question: What happens if another variable
is added in?
• Lets find out!
The relationship between AND
and OR
P(A or B)
P(A)
P(B)
P(A and B)
This is the relationship between AND and OR we
covered with two events. Now let’s see what it looks
like when you add in a third event.
With Three Events
P(A)
P(B)
P(C)
P(A) + P(B) + P(C)
1
2 1
2 3 2
1
Note- the numbers above correspond to how many times that area
of the figure is counted in the overall probability, which means if
all the numbers in the figure are 1,then it is an OR statement.
With Three Events
P(A) + P(B) + P(C)
1
P(A and C)
1
2 1
2 3 2
1
P(A and B)
P(B and C)
1 1
1 0 1
1
Note- the numbers above correspond to how many times that area
of the figure is counted in the overall probability, which means if
all the numbers in the figure are 1,then it is an OR statement.
With Three Events
P(A and B and C)
1
1
1 0 1
1
1
1
P(A or B or C)
1 1 1
1
1
1
1
Now that all the the numbers in the figure is one, it
is an OR statement, and for three variables.
With Three Events
So combining both steps from above, a formula to solve for the
P(A or B or C) would be:
P(A) + P(B) + P(C)
1
P(A and C)
P(A and B and C)
2 1
2 3 2
1
1
P(A and B)
P(B and C)
P( A or B or C)
With Three Events
Example Problem: The probability of Erik Johnson wearing a
green shirt (GS) is .34. The probability of him wearing a blue hat
(BH) is .12. The probability of him wearing orange socks (OS) is
.05. What is the probability of Erik wearing a green shirt, or a
blue hat or orange socks?
.34 x .05
= .017
.34 + .12 + .05
1
2 1
2 3 2
1
.34 x .12 x .05
= .00204
1
.34 x .12
= .0408
.12 x .05
= .006
.44824
2nd way to calculate OR with
Three Events
• Recall:
• P(A or B)= P(A) + P(A not occurring) x P(B)
• P(A or B or C)= P(A) +P(A not occurring) x
P(B) + what??
• Answer: Since another variable is being added
(C), you simply add P(A not occurring) x P(B
not occurring) x P(C) to the old equation.
Here’s the same sample problem as before calculated using
this method.
Example Problem: The probability of Erik Johnson wearing a
green shirt (GS) is .34. The probability of him wearing a blue hat
(BH) is .12. The probability of him wearing a orange socks (OS)
is .05. What is the probability of Erik wearing a green shirt, or a
blue hat or orange socks?
P(GS or BH or OS)=P(GS) +(P(GS not occurring) x P(BH)) +
(P(GS not occurring) x P(BH not occurring) x P(OS))
P(GS or BH or OS)=.34 + (1-.34) x .12 + (1-.34) x (1-.12) x
(.05)
P(GS or BH or OS)= .44824
• This concludes my presentation on two and
three events.
• Any Questions?