A question about the law of total probability

In summary, the question asks for the probability of two specific types of cheese, goat and parmesan, being on different sides of a randomly placed pickle among n different types of cheese. The solution involves using the law of total probability and defining the event Ai as the pickle being in the Ith space out of n-1 spaces. Another approach suggested is to first find the probability of goat and parmesan ending up together by combining them and then using that to determine the probability of them being on different sides.
  • #1
lola19991
6
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Dan put n different kinds of cheese in a row randomly, so that between two kinds of cheese there is a space. Then he puts a pickle on one of the n-1 spaces between the cheeses randomly. What is the probability of goat cheese and parmesan cheese (2 from n kinds of cheese) to be in the different sides of the pickle?

I know that I need to use law of total probability and define Ai as the pickle is found in the space number I from n-1 spaces, but I am stuck.
 
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  • #2
lola19991 said:
Dan put n different kinds of cheese in a row randomly, so that between two kinds of cheese there is a space. Then he puts a pickle on one of the n-1 spaces between the cheeses randomly. What is the probability of goat cheese and parmesan cheese (2 from n kinds of cheese) to be in the different sides of the pickle?

I know that I need to use law of total probability and define Ai as the pickle is found in the space number I from n-1 spaces, but I am stuck.

Don't we first need the probability that GOAT and PARM end up together? Perhaps what may seem like an odd approach, tape those two together and see if it is easier to count the possibilities.
 

FAQ: A question about the law of total probability

What is the law of total probability?

The law of total probability is a mathematical principle that states the total probability of an event occurring is equal to the sum of the probabilities of all possible outcomes of that event.

How is the law of total probability used in science?

In science, the law of total probability is used to calculate the probability of an event occurring in a complex system with multiple possible outcomes. It is often used in genetics, physics, and other fields to determine the likelihood of a particular outcome based on a set of known probabilities.

What is the difference between the law of total probability and the law of conditional probability?

The law of total probability deals with the total probability of an event occurring, while the law of conditional probability deals with the probability of an event occurring given that another event has already occurred. In other words, the law of total probability looks at all possible outcomes, while the law of conditional probability takes into account a specific condition.

Can the law of total probability be applied to both discrete and continuous variables?

Yes, the law of total probability can be applied to both discrete and continuous variables. It is a general principle that can be used in any situation where there are multiple possible outcomes.

What are some real-life examples of the law of total probability in action?

One example of the law of total probability in action is in predicting the outcome of a genetic cross. By knowing the probabilities of different genotypes, we can use the law of total probability to determine the likelihood of specific offspring. Another example is in predicting the outcome of a coin toss, where the law of total probability can be used to determine the likelihood of getting heads or tails based on the number of possible outcomes.

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