What is the Probability of making an even number of 4 digits using 1 2 3 repeated?

Answer

Hint: We use the principle of permutations and combinations to find the number of four digit even numbers using digits 1, 2, 3, 4 and 5. Thus, we use the fact that each digit has five possibilities and at the same time keeping in mind that we don’t repeat the digits. Further, to ensure that the number is even we ensure that in the unit’s place, the digits can only be 2 or 4.

Complete step-by-step answer:
First, since we have a constraint for the unit's place, we start by counting the combinations for the unit's place. Thus, we have only two possibilities – 2 or 4. Now, for the remaining places, we can start from any digit’s place to calculate the combinations. Suppose, we start with the tenths place, we now have five possibilities (1, 2, 3, 4 and 5) but we have to exclude one possibility since the unit’s place is already filled and no digit can be repeated. Thus, we would have four possibilities. Now, we move on to hundredths place, we would similarly have only 3 possibilities (since, now two places are filled and no digits can be repeated). Similarly, we would now have only two possibilities for the thousandths place. Thus, to calculate the total number of four digit even numbers, we multiply all possibilities to get the answer. Thus, we have, total possibilities-

2 $\times $ 4 $\times $ 3 $\times $ 2 = 48

Hence, there are 48 four digit even numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated.

Note: Generally, for questions regarding permutations and combinations involving calculations of number of digits, we start with the constraint portion of the problem. In this case, we start with the fact that we required the four digit number to be even and then proceed forward. This greatly helps us in simplifying the problem and further prevents from incorrectly counting the total numbers.

Learning Outcomes

• Compute a conditional probability for an event
• Use Baye’s theorem to compute a conditional probability
• Calculate the expected value of an event

We can use permutations and combinations to help us answer more complex probability questions.

examples

A 4 digit PIN number is selected. What is the probability that there are no repeated digits?

Try It

Example

In a certain state’s lottery, 48 balls numbered 1 through 48 are placed in a machine and six of them are drawn at random. If the six numbers drawn match the numbers that a player had chosen, the player wins $1,000,000.    In this lottery, the order the numbers are drawn in doesn’t matter. Compute the probability that you win the million-dollar prize if you purchase a single lottery ticket.

Example

In the state lottery from the previous example, if five of the six numbers drawn match the numbers that a player has chosen, the player wins a second prize of $1,000. Compute the probability that you win the second prize if you purchase a single lottery ticket.

The previous examples are worked in the following video.

examples

Compute the probability of randomly drawing five cards from a deck and getting exactly one Ace.

Example

Compute the probability of randomly drawing five cards from a deck and getting exactly two Aces.

View the following for further demonstration of these examples.

Try It

Birthday Problem

Let’s take a pause to consider a famous problem in probability theory:

Suppose you have a room full of 30 people. What is the probability that there is at least one shared birthday?

Take a guess at the answer to the above problem. Was your guess fairly low, like around 10%? That seems to be the intuitive answer (30/365, perhaps?). Let’s see if we should listen to our intuition. Let’s start with a simpler problem, however.

example

Suppose three people are in a room.  What is the probability that there is at least one shared birthday among these three people?

Suppose five people are in a room.  What is the probability that there is at least one shared birthday among these five people?

Suppose 30 people are in a room.  What is the probability that there is at least one shared birthday among these 30 people?

The birthday problem is examined in detail in the following.

If you like to bet, and if you can convince 30 people to reveal their birthdays, you might be able to win some money by betting a friend that there will be at least two people with the same birthday in the room anytime you are in a room of 30 or more people. (Of course, you would need to make sure your friend hasn’t studied probability!) You wouldn’t be guaranteed to win, but you should win more than half the time.

This is one of many results in probability theory that is counterintuitive; that is, it goes against our gut instincts.

Try It

Suppose 10 people are in a room. What is the probability that there is at least one shared birthday among these 10 people?

What is the Probability of making an even number of 4 digits using 1 2 3 and 4 without?

How many 4 digit even numbers can be formed using all the digits 1 2 3 4?

How many 4 digit even numbers can be formed using the digits 1 2 3 4 and 5 if your not allowed to repeat any digits?

How many even numbers of four digits can be formed with the digits 1 2 3 4 5 6 repetitions of digits are allowed )?