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Statistics 101: Principles of Statistics11 chapters | 144 lessons | 9 flashcard sets

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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!

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.

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!

Now Alex and Jon are playing a game that tests your reaction time. So far, Jon has won one game and Alex has won one game. There are three games left to play. In this game, Alex has an advantage. He has a better reaction time than Jon. Alex has a 60% chance over Jon of successfully capturing the light in between the two lines. Each guy will have 15 chances to capture the light. Alex has already played, capturing the light six times. What is the probability that Jon will win the game? Pause the video here to find the answer.

A little confused? Here's a hint: you'll have to do some calculations in order to find the information you need. First, if Alex has a 60% probability over Jon, then Jon's probability of capturing the light each time is 40%. Second, you need to figure out how many times Jon has to capture the light to win. Alex has captured the light six times, meaning Jon will need to capture the light seven times. So, there are your numbers: 40% chance of capturing the light, 15 tries and seven successes needed. Pause the video now to find the answer.

How did you do? Did you figure it out? The correct answer is approximately 18%. Let's see how we got there.

Here's the combination formula. The combination value is 6,435. Now, let's insert that and our other numbers into the binomial probability formula.

First, subtract the exponent 7 from 15 to get 8. Next, subtract .40 from 1 to get the probability of failure, which is .60. Solve each of the exponents: .40 to the seventh power is .0016384, and .60 to the eighth power is .01679616. Now, multiply from left to right to get a final answer of .1770, or approximately 18%. Yikes! It doesn't look good for Jon!

Remember, when working with the binomial probability formula it is important to make sure you have the correct information, break down the formula and follow the order of operations carefully. Here's another look at that formula.

Remember, that you'll 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. Also, you will need to find the combination for the binomial experiment.

You'll have to use the Internet or a graphing calculator to find the factorial values. With these formulas you'll be dealing with some serious multi-digit numbers, so don't forget to take it slow and don't make little mistakes! For more information about the binomial probability formula, check out our other lessons.

Each facet of the lesson is designed to prepare you to:

- Understand how to use the binomial probability formula
- Identify the values to be placed in the formula
- Utilize the Internet or a graphing calculator as you solve practice problems for probability

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Statistics 101: Principles of Statistics11 chapters | 144 lessons | 9 flashcard sets

- Go to Probability

- Random Variables: Definition, Types & Examples 9:53
- Finding & Interpreting the Expected Value of a Discrete Random Variable 5:25
- Developing Discrete Probability Distributions Theoretically & Finding Expected Values 9:21
- Developing Discrete Probability Distributions Empirically & Finding Expected Values 10:09
- Dice: Finding Expected Values of Games of Chance 13:36
- Blackjack: Finding Expected Values of Games of Chance with Cards 8:41
- Poker: Finding Expected Values of High Hands 9:38
- Poker: Finding Expected Values of Low Hands 8:38
- Lotteries: Finding Expected Values of Games of Chance 11:58
- Comparing Game Strategies Using Expected Values: Process & Examples 4:31
- How to Apply Discrete Probability Concepts to Problem Solving 7:35
- Binomial Experiments: Definition, Characteristics & Examples 4:46
- Finding Binomial Probabilities Using Formulas: Process & Examples 6:10
- Practice Problems for Finding Binomial Probabilities Using Formulas 7:15
- Mean & Standard Deviation of a Binomial Random Variable: Formula & Example 6:34
- Solving Problems with Binomial Experiments: Steps & Example 5:03
- Go to Discrete Probability Distributions

- Go to Sampling

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