একটি জটিল ঘটনার সম্ভাবনা

Suppose I have a bag of $10$ 3-dimensional shapes. There are $7$ cubes and $3$ prisms. The shapes are painted in the colors white and black. For every $10$ of these shapes, there will be $8$ white and $2$ black.

Let's talk about the probability that I pick a shape that's both a prism and it's painted white. This scenario is called compound probability, when there are two events which both have to be true. Right now, we have the two separate probabilities: the $P$(I get a prism) = $3/10$ and the $P$(it's white) = $4/5$.

Multiplying the two probabilities would be the best idea. I'll show you why. Pretend there's a family of kangaroos. Half of them are gray, and the other half are red. Now we can obviously see that the probability of a gray kangaroo is half. But that's not all. Out of the gray kangaroos, half are adults and half are babies. Same for the red ones. Do you agree that half of the kangaroos are adults? Well, half it is. Now if we are asking what fraction of all these kangaroos are red babies, half of them are red, and half of those red ones are babies, so we multiply these two fractions, obtaining one - fourth.

Well, if you've multiplied the correct fractions without a mistake, you would have gotten the answer $\frac{6}{25}$. Great job!

Probabilities of Compound Events

In statistics, the probability of compound events is the likelihood of $2$ events happening at once. Think of them as overlapping events.

More technically speaking, compound events are not mutually exclusive. This means they can both happen at the same time.

How'd we get that answer?

When calculating the probabilities of compound events, multiply the probabilities of the individual events together.

(Video) Probability of Compound Events

by Wetrodent

Wetrodent has a great video explaining the probability of compound events and explaining the difference between independent and dependent events.

$\large{\text{Summary}}$

This video reminds us that Compound Events consist of $2$ or more events. Events can either be dependent or independent, meaning their results are affected by one another, or their results are not affected by one another. Independent Events

The probability of two independent events can be found by multiplying the probability of the first event, by the probability of the second event. This notation can be written as $P(A \space and \space B) = P(A) \times P(B)$. In the video, he uses an example of rolling two dice to calculate the probability of two independent events. Each die is its own entity, and the result on each die does not depend on the other die. He asks the question “What is the probability of getting a $6$ on both die?” We can totally do this. Since each die has a $1$ in $6$ probability of rolling a $6$, that means that for die $1$, our probability of rolling a $6$ is $\frac{1}{6}$, and for die 2, our probability of rolling a $6$ is $\frac{1}{6}$. To find the total probability of having two $6$s is the result of multiplying the two together, or: $$\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$$

Dependent Events

If two events, $A$ and $B$, are dependent, then the probability of both events occurring is the product of the probability of $A$ and the probability of $B$ after $A$ occurs. This notation can be written as $P(A \space and \space B) = P(A) \times P(B \space following \space A)$. The example here is picking candy from a a pile of $5$. There are $3$ chocolate candies, so chances of picking a chocolate candy on the first try is $\frac{3}{5}$. This person picked a chocolate candy, so now once that is eaten, the new chances of picking a chocolate candy have dropped to $\frac{2}{4}$. Since a chocolate candy was picked first, and not replaced, the probability of picking a chocolate candy on the second try depended on the results of the first pull. Multiplying $\frac{3}{5}$ and $\frac{2}{4}$ together will give a result of $\frac{6}{20}$, which is the probability of picking a chocolate candy and then another chocolate candy.