Probability Product and Sum Rules

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“This video. We are going to discuss the product and some rules these are two two basic but important probability rules with a lot of different applications as a brief the product rule is applied to determine the joint probability of two or more independent events. Whereas. The set rule is applied to determine the total probability of two or more mutually.

Exclusive events. Each of these words will be explained in the next. Few slides. Let s start by looking at the product rule.

If a and b are two independent events then the probability of both a and b occurring is the product of the individual probabilities for a and b. In other words. The probability of a and b. Is the probability of a multiplied by the probability of b.

Independent. Events. Mean that the occurrence of one event does not affect the occurrence of the other so for example. If you were to toss a coin twice then getting ahead in the second toss is independent of what you got in the first coin toss the product rule.

Can also be used for multiple independent events in which case you would just multiply the probability of each of the independent events to get the total probability here s an example suppose that a randomly selected couple has two children of different ages. What is the probability that the first child is a boy and the second is a girl well giving birth to a boy and to a girl are equally likely in other words. The probabilities are the same in other words. A half each the gender of each tile is independent of the gender of the other child this means that we can apply the product rule.

So that the probability of first child being a boy and the second child being a girl is the probability of the first child being a boy multiplied by the second child being a girl in other words half multiplied by 1 2. Which gives the quarter now to some rule. If a and b are two mutually exclusive events then the probability of either one or the other recurring is the sum of the individual probabilities for a and b. In other words.

The probability of a or b. Is the same as the probability of a plus. The probability of b. Mutually.

Exclusive events. Means that the events could not both occur at the same time for example. Monday and tuesday will be mutually exclusive. But monday in april would not be the same rule can also be applied.

If you have several events as long as they are all mutually exclusive in that case just add each of the different mutually exclusive events to get the total probability of one of them occurring. Here s an example of the sum rule. Suppose that in country x..

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47. Of the population has blood type. A and 20 has blood type b. What is the proportion of the total population that has either blood type a or blood type b.

Well. Each person can only have one blood type this means that having blood type. A and having blood type b. Are mutually exclusive events this means that we can use the sum rule in other words.

The total probability is given by adding the two different probabilities. Forty seven percent plus twenty percent is 67. Percent. Now.

Let s have a look at a longer example that involves both the product and the sum rules. Suppose the kate and john have these given genotypes in three genes where capital letter. Represents a dominant allele and a lowercase letter represents. A recessive allele.

What is the probability that their child will possess all three dominant traits to have all three dominant traits their child must have a phenotype that involves at least one dominant allele in each of the three genes in other words p. Dash. Could represent big p. Big p.

Or big. P. Small p. And similarly for the others.

Each gene is independent of the others. So this means that we can use the product rule. In other words to get the total probability we need to multiply three independent probabilities together this means that to solve this question we need to find each of these three independent probabilities. And it s final step multiply them together first let s find the probability of t.

Remember that k test unit. I big p big p. And john has genotype small p. Small p.

We can fill these out in a punnett square and fill out the different possibilities for their child. But notice that there is only one genotype possible for the child big p. Small p..

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This means that the probability of p. Is the same as the probability for a big p. Small p. Which is equal to one in other words 100.

So that was easy. Now. Let s have a look at the probability. For big.

Q. Remember that k test. Unit. Type.

Small. Q. Small. Q.

And john has genotype big. Q. Small q. Again.

Let s fill out the different possibilities for their child in a punnett square. This time there are two different possibilities since the child can only get small q from kate s this means that the probability of this is 100. However from john it could get us big q. Or a small q.

With equal probabilities. So each of these has probability 1 2. The allele that is received from john is independent of that received from kate s in other words. We should use the product rule so to get the probability of q.

We will find the probability of big q. Small q. Which is the probability of small q from kate multiplied by the probability of q. From john in other words.

1. Multiplied. By 1 2..

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Which is equal to 1 2. Now let s find the probability of r. Each parents. Have the same genotype.

Big r. Small r. So this time filling out the punnett square. We see that there are three possible genotypes for the child.

Let s try to find the probability of each of these three different types notice first that from each parent. Leaving a bigger or a smaller is equally likely in other words each of them have probability 1 2. Which we can fill out in the table. Also the allele that is received from cates is independent of that from john in other words.

We should use the product rule. If the child is going to have genotype. Big r. Big r.

This means that it must have received the pick r from each of the two different parents because these are independent of each other this means that we should multiply the two different probabilities and 1 2. Times. 1 2. Is equal to 1 4.

Similarly. The probability of getting a big r from the mother. And a small art from the father is the same as big r from the father and small art. From the mother and each of these is 1 2.

Times 1 2. Which is equal to 1 4. So we can fill all these out. In this table.

And also similarly small r. Small r. Is also 1 4. Now that we have filled out all the probabilities.

In this table. We can put this together and find the probability of r. The probability of getting a big r from the mother and the smaller from the father compared to the other way around are mutually exclusive events..

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Because they cannot both happen in the same child this means that we can use the sum rule. In other words to find the probability of big r. Small r. We should add 1 4.

Plus. 1. 4. To get 1 2.

Similarly. The different genotypes are mutually exclusive events. Because they nut cannot happen in the same child. This.

Means that we can add. These probabilities the quarter plus 1. 2. Is equal to 3 4.

Now we can finally put all this together to answer the final question. Which was what is the probability that the child will possess all three dominant traits notice that so far. We ve worked on three completely separate steps to find these 3. Different probabilities.

And only now can we start putting this together the formation of the genotype for one gene is independent of the other genes this means that we can use the product rule. So to find the total probability. We should multiply the three probabilities that we found. In step.

1. 2. 3. 1.

Times 1 2. Times. Three quarters. Gives us three over eight.

And this is the final answer here s a summary with the question and the final answer and some of the important probabilities that we computed along the way. ” ..

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