4.3.2 Combined conditional probabilities
Combined conditional probabilities refer to the probabilities of multiple events occurring together, given that certain conditions are met.
These probabilities are calculated using conditional probability, which is the probability of an event occurring given that another event has already occurred.
For example, suppose we have events \( A \), \( B \), and \( C \).
The combined conditional probability of all three events occurring together, given that event \( D \) has already occurred, is denoted as \( P (A \ and \ B \ and \ C|D) \) and is calculated using the following formula:
\( P(D)=\frac{P(A \ and \ B \ and \ C \ and \ D)}{P(D)} \)
Where \( P (A \ and \ B \ and \ C \ and \ D) \) represents the probability of all four events occurring together
\( P(D) \) represents the probability of event \( D \) occurring.
Combined conditional probabilities are often used in fields such as finance, economics, and engineering, where it is necessary to consider the likelihood of multiple events occurring together, given certain conditions.
To solve combined conditional probability questions
Identify the events and their probabilities:
Determine the events involved in the problem and their respective probabilities.
Make sure to pay attention to whether the events are independent or dependent.
Apply the appropriate formula:
Use the multiplication, addition, conditional probability, or total probability formula depending on the problem.
Give a quick look into the formulas
Multiplication Rule:
\( P(A \ and \ B)=P(A)\times P(B|A) \)
This formula is used when we want to find the probability of two events \( A \) and \( B \) occurring together.
The probability of \( B \) occurring is conditional on \( A \) already having occurred.
Addition Rule:
\( P(A \ or \ B)=P(A)+P(B)-P(A \ and \ B) \)
This formula is used when we want to find the probability of either event A or event B occurring.
If the events are not mutually exclusive, we need to subtract the probability of their intersection to avoid double-counting.
Conditional Probability Formula:
\( P(A|B)=\frac{P(A \ and \ B)}{P(B)} \)
This formula is used when we want to find the probability of event \( A \) occurring given that event \( B \) has already occurred.
We divide the probability of both events occurring by the probability of event \( B \) occurring.
Total Probability Formula:
\( P(A)=∑ P(A|B)\times P(B) \)
This formula is used when we want to find the probability of event \( A \) occurring, taking into account all possible ways it can occur, each with its own probability.
We sum the product of the probability of \( A \) given \( B \) and the probability of \( B \) over all possible values of \( B \).
Simplify the expression:
Simplify the expression by substituting the given probabilities and calculating any necessary products, sums, or conditional probabilities.
Solve for the desired probability:
Use algebraic techniques to isolate the probability you are trying to find.
Check your answer:
Make sure your answer makes sense and that it is consistent with the problem statement.
It is important to carefully read and understand the problem, and to draw a diagram or table if necessary to keep track of the probabilities and events involved.
It is also important to pay attention to the wording of the problem and to make sure you are using the correct formulas and assumptions.
Worked example:
A box contains \( 7 \) black pens and \( 8 \) orange pens.
Two pens are chosen at random from this box without replacement.
Calculate the probability that at least one orange pen is chosen.
Find the probability that both pens are black.
\( P(both \ are \ black)=\frac{7}{15}\times \frac{6}{14}=\frac{42}{210} \)
The only way that at least one pen isn’t orange is if both pens are black
So
\( P(at \ least \ one \ pen \ isn’t \ orange)=1-P(both \ pens \ are \ black) \)
\( P(at \ least \ one \ pen \ isn’t \ orange)=1-\frac{42}{210}=\frac{168}{210} \)
\( \frac{168}{210} \)
Test yourself
Question 1:
A bag contains four red marbles and two yellow marbles.
Behnaz picks two marbles at random without replacement.
Find the probability that
the marbles are both red,
[2]
the marbles are not both red.
[1]
Question 2:
Bag \( A \) contains \( 3 \) black balls and \( 2 \) white balls.
Bag \( B \) contains \( 1 \) black balls and \( 3 \) white balls.
A ball is taken at random from each bag.
Show that a black ball is more likely to be taken from bag \( A \) than from bag \( B \).
[1]
Find the probability that the two balls have different color.
[3]
The balls are returned to their original bags.
Three balls are taken at random from bag \( A \), without replacement.
Find the probability that they are all black.
[2]
Find the probability that they are all white.
[1]
The balls are returned to their original bags.
A ball is taken at random from bag \( A \) and its color is recorded.
This ball is then placed in bag \( B \).
A ball is then taken at random from bag \( B \).
Find the probability that the ball taken from bag \( B \) has a different color to the ball taken
from bag \( A \).
[3]
Question 3:
This year, \( 40 \) students have each traveled by one or more of plane \( P \), train \( T \) or boat \( B \).
\( 7 \) have traveled only by plane.
\( 11 \) have traveled only by train.
\( 9 \) have traveled only by boat.
\( nP \) ∩ \( T =8 \)
\( nB \) ∩ \( T=3 \)
\( nB \) ∩ \( P=6 \)
Complete the Venn diagram.
[3]
Two students are chosen at random .
Calculate the probability that they both have traveled only by plane.
[2]
Two students are chosen at random who have traveled by train.
Calculate the probability that they both have also traveled by plane.
[2]
Question 4:
Angelo has a bag containing \( 3 \) white counters and \( x \) black counters.
He takes two counters at random from the bag, without replacement.
Complete the following statement.
The probability that Angelo takes two black counters is
\( \frac{x}{x+3}=………………. \)
[2]
The probability that Angelo takes two black counters is \( \frac{7}{15} \).
Show that \( 4x^2-25x-21=0 \).
[4]
Solve by factorisation \( 4x^2-25x-21=0 \)
[3]
Write down the number of black counters in the bag.
[1]
Question 5:
The diagram shows two sets of cards.
Jojo chooses two cards at random from set \( A \) without replacement.
Find the probability that the two cards have the same number.
[3]
Jojo replaces the two cards.
Kylie then chooses one card at random from set \( A \) and one card at random from set \( B \).
Find the probability that the two cards have the same number.
[3]
Who is the most likely to choose the two cards that have the same number?
Show all your working.
[1]
Lena chooses three cards at random from set \( C \) without replacement.
Find the probability that the third card chosen is numbered \( 4 \).
[3]