Accelerated Geometry Probability Probability Test Review Conditional Probability
Provisional Probability
Allow'south look at conditional probability, with some calculations and examples.
Provisional probability tin be defined as the likelihood of an event B happening given the fact that an outcome A has already happened. It means that upshot B is dependent on issue A, or that event A is a condition for outcome B to happen.
How to calculate provisional probability
In that location is a formula nosotros can utilise to find the probability of B given A has already occurred:
Where:
P(B|A) is the probability of B given A
P(A∩B) is the probability of both A and B occurring and
P(A) is the probability of A occurring
In an international school, there are 32 pupils in i particular class. five of them are Italian. 3 of the Italian students are boys. A educatee is picked at random from the grade. What is the probability of that student being a boy given that the student is Italian?
In this case, nosotros are looking for P(boy | Italian). Using the formula above,
P(boy|italian)= 
The probability that a pupil is Italian and is a boy is 
= 0.09375
The probability that a pupil is Italian is 
= 0.15635
Therefore, the probability that a student is a male child and is Italian is:
P(boy|italian)= 
=0.6 or sixty%.
Conditional probability tree diagram
A Tree Diagram can exist a useful style to visualise and solve problems that contain conditional probabilities. What nosotros need to do is depict the starting time two branches for event A and then the 4 branches for event B.
For instance, allow'due south imagine we had a purse containing 10 sweets that were either strawberry or lemon flavoured. We then picked one random sweet from the handbag, ate it and then picked another ane. If we knew that in that location were six strawberry sweets at the start, we could kickoff cartoon a tree diagram showing the probabilities of picking either lemon or strawberry sweets. The first fourth dimension nosotros picked, the probability of picking a strawberry sweetness would be six/x or 0.6 and hence the probability of picking a lemon sweetness has to be 1-0.6=0.4. From this, we can draw the kickoff branches of our tree diagram.
First branches of tree diagram showing the probability of picking a strawberry or lemon sweetness from a pocketbook
Now, what happens for the 2nd sweet we selection? Call up that the first sweet nosotros picked was not put back into the handbag and so the total number of sweets in the bag is now 9 and the flavour picked on the second draw is dependent on the flavor picked in the first depict . If on the first selection, nosotros took a strawberry sweetness, in that location will simply be 5 strawberry sweets left in the pocketbook so the probability of picking a strawberry sugariness now would be 5/nine=0.556 and the probability of picking a lemon sweet would be 1 - 0.556= 0.444
However, if in the start pick we took a lemon sweetness, there volition now be six strawberry and iii lemon sweets left. And so the probability of picking a strawberry sweetness in this situation is half dozen/9 =0.667 and the probability of picking a lemon sugariness is 1-0.667 = 0.333. We tin can at present draw the next four branches in our tree diagram:
Tree diagram showing the probability of picking 2 sweets from a handbag containing strawberry and lemon sweets
The four branches we drew correspond the provisional probabilities of different events. The commencement i from the top gives the probability of picking strawberry (S) given that strawberry was already picked the kickoff time, then 
. This aforementioned logic can exist applied to all the post-obit branches, giving
and 
.
Provisional probability Venn diagram
Venn diagrams are another method we can use to solve provisional probability problems. To draw a Venn diagram, we need to know the probability of issue A, the probability of event B and the probability of A and B. For example, a survey was conducted on 65 people asking about the flavours of water ice foam they liked. 30 people said they similar only chocolate (C), twenty people said they liked just vanilla (V), 10 people liked both and the rest liked neither. From this, we know that the probability that someone simply likes chocolate is 30/65=0.462. The probability that someone likes both vanilla and chocolate is P(C∩V)=ten/65=0.154. The probability that someone likes neither is 65-(thirty+20+x)/65 = 0.0769. Nosotros tin draw the following Venn diagram:
Venn diagram showing the probability of people liking chocolate and vanilla ice cream.
Notation that adding all the probabilities on a Venn diagram should always equal one.
From this, we tin can utilize the formula 
to find the probability that someone likes chocolate ice cream given that they like vanilla:
.
We already know that P(C∩Five)= 0.154. P(V) will be the sum of the probabilities of someone liking just vanilla and someone likes both vanilla and chocolate. P(V)=0.308+0.154=0.462. And so,
.
The probability that Tom starts to smoke is 0.2. The probability that he starts smoking and then develops lung cancer is 0.15. What is the probability that he develops lung cancer given that he started smoking?
From this, we already know that P(A)=0.ii and 
. The question asks for the probability that Tom develops lung cancer given that he has started smoking, and so that is 
. Using the provisional probability formula, we tin can work out that 
. So the probability that Tom develops lung cancer if he starts smoking is 75%.
Bayes theorem for conditional probability
Bayes theorem states that 
This theorem tin can be proved by using the equations we used before. We know that 
and 
.
If nosotros rearrange both equations to find expressions for 
, we go 
and 
. We can now equate the right-hand side of both of these expressions, then 
, therefore, 
l% of rainy days start off cloudy. 40% of all days start cloudy. In a particular calendar month, it rains 3 out of 30 days. What is the probability that for a day during this calendar month it rains given that the solar day starts cloudy?
We know that 
and 
From Bayes theorem, nosotros know that 
. Therefore,
Then the probability that it rains given that information technology's cloudy is 12.5%.
Conditional Probability - Key takeaways
-
Conditional probability is the probability of an event B occurring given that some other event A has already occurred.
-
In provisional probability, issue B is dependent on event A
-
The formula for the probability of B occurring given A is
-
Tree diagrams and Venn diagrams can also be used to find conditional probability
-
Bayes theorem states that
Conditional Probability
Conditional probability is the probability of an outcome B occurring given that some other event A has already occurred.
No, conditional probability tin can never be greater than 1. In fact, no probability can ever be greater than 1.
Provisional probability can be calculated using the post-obit formula: P(B|A)=P(A∩B)/P(A)
Final Conditional Probability Quiz
Source: https://www.studysmarter.de/en/explanations/math/statistics/conditional-probability/
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