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It is quite easy to calculate simple probabilities. What is the chance of rolling a 4 with a single die? What is the chance of being dealt a queen from a deck of cards? We are now going to focus on conditional probabilities. A conditional probability is a probability in which a certain prerequisite condition has already been met.
We can start by thinking about cards being dealt from a standard deck of 52 cards. As each card is dealt, what remains in the deck changes. A gambler in a casino will pay close attention to cards played. If many face cards have already been dealt, the observant gambler will understand that the next card has a higher chance of not being a face card. Suppose we want to know the probability that our next card will be a face card given that the first card was the 7 of diamonds. The formal notation for this is P(Face|7♦). This is read as "The probability of a face card given that we already have been dealt the 7 of diamonds.". Often times the math for these situations is very logical. In our case, we have simply reduced the deck by one card and there are still 12 face cards in the deck. Therefore P(Face|7♦)=
.
Example 1
Two cards are dealt from a standard deck of 52 cards. Find each conditional probability.
a) P(2nd red|1st 2♣)
b) P(2nd red|1st 2♦)
c) P(2nd club|1st red)
Solution
a) There are still 26 red cards out of the remaining 51 cards after the two of clubs is dealt. .
b) There are only 25 red cards out of the remaining 51 cards after the two of diamonds is dealt. .
c) There are still 13 clubs out of the remaining 51 cards after a red card is dealt. .
Example 2
In a common poker game, 5 cards are dealt to a player. The best possible hand is called a royal flush. This occurs if a player gets the ten, jack, queen, king, and ace all of the same suit. What is the chance of being dealt a royal flush? Leave your answer as a fraction.
Solution
We will solve this by looking at one card at a time. What is the chance that the first card might be part of a royal flush? Before any cards are dealt, there are four 10's, four jack's, four queen's, four kings, and four aces available. Twenty of the 52 cards can help you on your way to a royal flush.
Once you receive this card, what are the chances that the second card will also help on your way to the royal flush? We might answer this by simply imagining us getting a useful card on the first card. Suppose our first card was the jack of spades. There are only 4 other cards of the remaining 51 cards that will help now, the 10, queen, king, and ace of spades. Suppose we get one of those cards, perhaps the king of spades.
There are now only 3 cards of the remaining 50 that can help us complete our royal flush. Suppose our third card was the queen of spades. Only 2 of the remaining 49 cards will help our quest for our fourth card. Likewise, there is only one card of the last 48 that can help us on card number five. Putting this all together, we have
Putting this in perspective, if you dealt 1000 poker hands every single day, it would take nearly two years to deal 649,740 hands, of which we would only expect about one to be a royal flush. Good Luck!
Another way we can look at conditional probabilities is through the use of two-way tables or contingency tables. These are often referred to as two-way tables because there are two distinct pieces of information gathered in these tables. For example, we may record how many siblings you have and in how many activities you participate in school. Two-way tables can be filled in either using counts or probabilities.
We will start by answering simple questions such as "What is the probability that a student participates in exactly 2 activities?". After we understand how to work with these tables, we will begin asking more complex questions such as "What is the chance a student participates in 3 activities given that they have 1 sibling?". Let's begin with an easy example to help us understand how to read these tables.
Example 3
Suppose we survey all the students at school and ask them how they get to school and also what grade they are in. The chart below gives the results. Suppose we randomly select one student.
[Figure3]
a) Give all the row and column totals.
b) What is the probability that the student walked to school?
c) What is the probability that the student was a 9th or 10th grader?
d) What is the probability that a student either rode the bus or is in 11th or 12th grade?
Solution
a)
b) There were 88 walkers out of 500 total students or .
c) There were 210 9th or 10th graders out of 500 total students or .
d) There are 147 kids who rode the bus and there are 290 kids who are 11th or 12th graders. However, notice that these two categories intersect and we must be careful not to count the 41 kids who are in both categories twice. We will take the 290 11th or 12th graders and just add the 106 bus riders who are not 11th and 12th graders for a total of 396 students. The probability of selecting an 11th or 12th grader or a bus rider is
.
In the example above, note that the total across the bottom, 147+88+254+11, and the total for the last column, 210+290, both add up to 500. This is true of all 2-way tables. Now that we have the basic ideas down in a contingency table, let's move to a couple of more challenging questions.
Example 4
Consider the completed chart in the solution of part a) of Example 3.
a) What is the probability that a student is in 11th or 12th grade given that they rode in a car to school?
b) What is P(Walk|9th or 10th grade)?
Solution
a) The trick to dealing with conditional probabilities in two-way tables is to make sure that you only use what you are given. We are given that they rode in a car to school. We will only look at the Car column. We first note that there were a total of 254 kids who rode in a car to school. We then see that 184 of these kids were 11th and 12th graders. This gives us
.
b) We want the probability that a student walked to school given that they were in 9th or 10th. We will only look only at the 9th and 10th grade row. There are 210 students who are 9th and 10th graders. Of these, only 30 walked to school. This gives us
.
Example 5
The manager of an ice cream shop is curious as to which customers are buying certain flavors of ice cream. He decides to track whether the customer is an adult or a child and whether they order vanilla ice cream or chocolate ice cream. He finds that of his 224 customers in one week that 146 ordered chocolate. He also finds that 52 of his 93 adult customers ordered vanilla. Build a contingency table that tracks the type of customer and type of ice cream.
Solution
Start by filling in the values we are given and then work from there. The table below shows what we are given in the initial problem.
Our next step is to fill in the Adult/Chocolate space with 41, the Child/Total box with 131, and the Total/Vanilla box with 78 by using subtraction. It is now easy to fill in the remaining boxes. For example, we can quickly determine that the Child/Chocolate box must be 146 - 41 = 105. You can verify that this table is correct by checking each row and column total.
Example 6
A survey asked students which types of music they listen to? Out of 200 students, 75 indicated pop music and 45 indicated country music with 22 of these students indicating they listened to both. Use a Venn diagram to find the probability that a randomly selected student listens to pop music given that they listen country music.
Solution
Consider our Venn diagram below. First fill in the both section with 22 students. We can now logically deduce how many students are left to fill up the Pop circle and the Country circle. Since we are given that they listen to country music we may only use the information that is in the Country circle. There are only 45 students that landed in this circle. Of the 45 students who listen to country music, 22 of them also listen to pop music or
.
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