Can someone explain to me the difference between joint probability distribution and conditional probability distribution?
$\endgroup$3 Answers
$\begingroup$Broadly speaking, joint probability is the probability of two things* happening together: e.g., the probability that I wash my car, and it rains. Conditional probability is the probability of one thing happening, given that the other thing happens: e.g., the probability that, given that I wash my car, it rains.
Consider the space of all four combinations:
- I wash my car, and it rains.
- I wash my car, and it doesn't rain.
- I don't wash my car, and it rains.
- I don't wash my car, and it doesn't rain.
Quantitatively, the difference is that in the case of joint probability, we evaluate across the space of all four combinations: The joint probability is the the first, divided by the sum of all four. (Note: If we're talking about a probability space, the sum of all four is everything—i.e., it has probability one.)
In the case of conditional probability, we only examine those cases where I wash my car (the antecedent). So the conditional probability is the first, divided by the sum of the first two.
*More than two things could be involved. But the basic idea can most clearly be illustrated with two things.
$\endgroup$ $\begingroup$I will answer this question with a typical example as it might be easier to understand the difference, but first for some definitions:
Definition 1:
Given two independent events $A$ and $B$ $$P(A \space\mathrm{and}\space B)=P(A)\times P(B)=\space$$The probability of event $A$ and event $B$ happening simultaneously. This is known as $\color{#180}{\fbox{Joint Probability}}$
Definition 2:
$$P(A\mid B)=\cfrac{P(A \space\mathrm{and}\space B)}{P(B)}=\space$$The probability that event $A$ occurs given that event $B$ has occurred. This is $\color{#F80}{\fbox{Conditional Probability}}$.
The following table gives data on the type of car, grouped by petrol consumption, owned by $100$ people.
$$\begin{array}{|c|l:l|}\hline & \mathsf{Low} & \,\mathsf{Medium} & \,\mathsf{High}& \,\mathbf{Total} \\ \hline \mathsf{Male} & 12 & 33 & 7 \\ \hdashline \mathsf{Female} & 23 & 21 & 4 \\ \hdashline & & & & 100 \\ \hline \end{array}$$
One person is selected at random.
Let $L$ be the event "the person owns a low rated car"
Let $F$ be the event "a female is chosen".
Questions:
Find
i). $P(L)=\space$The probability of selecting someone with a low rated car,
ii). $P(F \space\mathrm{and}\space L)=\space$The probability of selecting a female with a low rated car.
iii). $P(F\mid L)=\cfrac{P(F \space\mathrm{and}\space L)}{P(L)}=\space$The probability of selecting a female given the car is low rated.
Solutions:
i).
$$\begin{array}{|c|l:l|}\hline & \mathsf{Low} & \,\mathsf{Medium} & \,\mathsf{High}& \,\mathbf{Total} \\ \hline \mathsf{Male} & \color{blue}{12} & 33 & 7 \\ \hdashline \mathsf{Female} & \color{blue}{23} & 21 & 4 \\ \hdashline & \color{red}{35}& & & \color{red}{100} \\ \hline \end{array}$$ $$P(L)=\frac{12+23}{100}=\frac{35}{100}=\frac{7}{20}$$
ii). $$\begin{array}{|c|l:l|}\hline & \mathsf{Low} & \,\mathsf{Medium} & \,\mathsf{High}& \,\mathbf{Total} \\ \hline \mathsf{Male} & 12 & 33 & 7 \\ \hdashline \mathsf{Female} & \color{red}{23} & 21 & 4 \\ \hdashline & & & & \color{red}{100} \\ \hline \end{array}$$ $$P(F \space\mathrm{and}\space L) = \cfrac{23}{100}$$ This is the joint probability and is the probability that two events happen simultaneously, note that this is sometimes written $P(F \cap L)$ which is known as the intersection of events $F$ and $L$.
iii). We need to be careful with the denominators in ii). and iii). Here we are given that the car is low rated; so we want the total of that column: $$\begin{array}{|c|l:l|}\hline & \mathsf{Low} & \,\mathsf{Medium} & \,\mathsf{High}& \,\mathbf{Total} \\ \hline \mathsf{Male} & \color{blue}{12} & 33 & 7 \\ \hdashline \mathsf{Female} & \color{blue}{23} & 21 & 4 \\ \hdashline & \color{red}{35} & & & 100 \\ \hline \end{array}$$
$P(F\mid L)=\cfrac{23}{35}$
It's helpful to notice that from the definition of conditional probability: $P(L)\times P(F\mid L)=\cfrac{7}{20}\times\cfrac{23}{35}=\cfrac{23}{100}=P(F \space\mathrm{and}\space L)$ as calculated in part ii).
Just one small note: When I put "events $A$ and $B$ happen simultaneously" the word 'simultaneously' is not essentially describing time; it is just a word we use to state that both events happen at once.
$\endgroup$ 2 $\begingroup$How can you apply the first equation in the above, i.e. $P(F \, \text{and} \, L)=P(F)*P(L)$? $P(F)=12/25$, $P(L)=7/20$, then the product is $28/125$ which is different from $23/100$ ? What mistake did I make here?
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