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Practice Problems for Finding Binomial Probabilities Using Formulas

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  • 0:01 Understanding Binomial…
  • 2:49 Practice Problem 1
  • 4:11 Practice Problem 2
  • 6:20 Lesson Summary
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Lesson Transcript
Instructor: Cathryn Jackson

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

In this lesson, you will review the binomial probability formula and the combination formula. Then follow along with the practice problems and see if you've mastered these concepts!

Understanding Binomial Probabilities

Alex and Jon are at their favorite arcade. Jon wants to make a bet: $20 that he will win three out of the next five games they play. Assuming that they are evenly matched and that there is a 50% chance that either one of them could win, what is the probability that Jon will win the bet?

To solve this problem you need to understand how to use a binomial probability formula. Since this is a practice problem lesson, let's quickly review how to solve a problem using Jon's bet as an example.

First, we need to find the values of x, n and P. The x represents the number of successes, the n represents the number of trials and the P represents the probability of success on an individual trial. In our case, x represents the number of games that Jon wins. Jon bets he can win three out of five games, therefore x = 3. The n represents the number of games Jon and Alex will play, therefore n = 5. Finally, The P represents the probability of an individual trial. Jon and Alex are evenly matched, therefore P = .50

When we plug our numbers into the formula it should look like this. But don't forget about C! The formula for C looks like this. Remember to use a graphing calculator to find the factorials in this formula. If you have trouble, make sure you break it down and check that you are using order of operations properly.

The combination is 10 for this particular problem. Now that we know this number, let's insert it into our binomial probability formula.

Okay, so for this problem, I started out by subtracting the exponents of 5 and 3 to get 2, which means Jon will have to lose two games and win three in this scenario. I also inserted our combination number, 10, into the equation. Next, on the fourth row, I subtracted .50 from 1. This gets us the probability of failing a single trial (or losing a game), and since Jon and Alex are equally matched, the probability is still .5. In the fifth row, I calculated .5 to the third power, and in the sixth, I calculated .5 to the second power. Lastly, I multiplied from left to right, which led me to the answer .3125. This means that Jon has a 31% chance that he will win only three out of the five games, no more and no less.

Now that we've reviewed how to use the binomial probability formula, let's look at some other practice problems.

Practice Problem 1

The first game Alex and Jon decide to play is a simple basketball game. A hoop is placed in the machine, and basketballs are dispensed to the players. The person to make the most baskets wins. Alex and Jon are once again evenly matched for this game. They decide to each shoot ten baskets. Jon will need to make six baskets to win the game. What is the probability that Jon will make only six baskets, no more and no less? Pause the video here to find the answer.

How did you do? The correct answer is .2050, or approximately 21%. Let's break down this problem.

First, you need to calculate the combination. I got 210. Now let's plug this number into our binomial probability formula.

Once again I insert the combination number, and I subtract the exponents 10 - 6. Next I find the probability of failure, which is .5. In the fifth row, I take .50 to the sixth power, which gets me .015625. Next, I calculate .5 to the fourth power, which is .0625. Last, I multiply from left to right and end up with .2050, which is approximately 21%. Let's try another problem!

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