PS - Finding Probabilities Lesson

Finding Probabilities

As discussed in the introduction, finding the probability of simple events can be exactly that, simple. The wrinkles come in when we are finding the probability of more than one thing occurring. We then have to decide if one event affects the other. If so, we have to take that into account. We will begin with some simple probabilities to get warmed up.

Probabilities of Simple Events

Simple events are those discussed earlier. They are the probability of one thing happening, such as the probability of rolling a four on a six sided die or the probability of a flipped coin coming up "heads". These can be calculated by dividing the number of favorable outcomes by the total number of outcomes.

$P\left(4\right)=\frac{number\:of\:fours}{number\:of\:sides\:of\:the\:die}=\frac{1}{6}\:\:\:\:\:\:\:\:\:\:\:P\left(Heads\right)=\frac{number\:of\:heads}{number\:of\:sides\:of\:a\:coin}=\frac{1}{2}$

Probabilities of Compound Events

Compound events are events where more than one thing must occur. In this case, we have to decide if one outcome will affect the other. This is called dependence. We will discuss dependence and independence in more detail in the next lesson. In this lesson, we will focus on independent events. These probabilities are easier to calculate.

The best way to understand compound events is to think of a tree diagram. Let's look at a tree diagram that combines the two situations above. If a coin is tossed and a die is rolled, what is the probability of getting heads and a 4?

The probability of flipping "Heads" is 1/2. If you were to flip "Heads", when you roll the die, you could roll any number from 1-6. So after flipping heads, you have a 1/6 probability of rolling a 4. So you only roll a four 1/6 of the times you flip "Heads". This means the probability of doing both is 1/6 of 1/2. This makes the total probability is 1/12.  As you can see in the tree diagram, the outcome of rolling a 4 after flipping "Heads" is one outcome out of a total of 12.

The simple way of calculating this is to start with the first probability (1/2) and multiply it by the second probability (1/6). This will give you the resulting probability of 1/12. This gives us the multiplication rule for independent events. The probability of A and B occurring is the probability of A times the probability of B.

Now you can practice using the Multiplication Rule for Independent Events. While playing a game, a player must roll a 6 sided die and then a 10 sided die. Find the probability of each of the following outcomes.

Now take a look at this presentation explaining how intersections affect our probabilities.

Conditional Probability

Conditional probability is the probability of an event given that another event has occurred. An easy way to see this is to look at the events in a two-way table.

	Yes	No	Total
Upper-classmen	300	100	400
Lower-classmen	450	150	600
Total	750	250	1000

Here is our vending poll data from earlier. An example of conditional probability would be, "What is the probability that someone answered yes given that they are an upper-classman?" In this case, we already know that the person is an upper-classman and want to know the probability that they said "yes". The notation looks like this: P(Yes|Upper)

Since we know the person is an upper-classman, we only care about the upper-classmen who answered the poll.

	Yes	No	Total
Upper-classmen	300	100	400

Looking at the Upper-Classmen, we can see that 300 out of 400 students said "yes". This means:

$P\left(Yes|Upper\right)=\frac{300}{400}=\frac{3}{4}$  

This is different than if we ask, "What is the probability of someone being an Upper-Classmen given they said yes?"

P(Upper|Yes)

Now we only care about those who answered "yes" and we are looking for Upper-Classmen.

Yes
300
450
750

You can see that of all 750 people who said "yes", 300 were Upper-Classmen.

$P\left(Upper|Yes\right)=\frac{300}{750}=\frac{2}{5}$

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