# Chapter 1. Normal Approximation to Binomial Distributions

## Introduction

Statistical Applets

You can use the sliders to change both n and p. Click and drag a slider with the mouse. Start by choosing p. The binomial distributions are symmetric for p = 0.5. They become more skewed as p moves away from 0.5. The bars show the binomial probabilities. The vertical gray line marks the mean np. The red curve is the normal density curve with the same mean and standard deviation as the binomial distribution. As you increase n, the binomial probability histogram looks more and more like the normal curve.

Click the "Quiz Me" button to complete the activity.

The Central Limit Theorem says that as n increases, the binomial distribution with n trials and probability p of success gets closer and closer to a normal distribution. That is, the binomial probability of any event gets closer and closer to the normal probability of the same event. The normal distribution has the same mean μ = np and standard deviation as the binomial distribution.

### Question 1.1

Suppose you flip a "fair" coin (that is, one with probability 0.5 of coming up either heads or tails) 16 times. According to the applet, the most likely result will be that L6bSXEGJIC8= of the tosses will come up heads. The probability of this outcome, according to the applet, is approximately eVl1o4RLWZscZlBuQFUlkYJjl4TkN8LtG1MQOqnfHmc=.

2
Try again.
Incorrect. See above for the correct answers.
Great job.
Suppose you flip a "fair" coin (that is, one with probability 0.5 of coming up either heads or tails) 16 times. According to the applet, the most likely result will be that ____ of the tosses will come up heads. The probability of this outcome, according to the applet, is approximately ____.

### Question 1.2

Now increase the number of trials to 50. According to the applet, the most likely result will be that Bqf/EtUReUc= of the tosses will come up heads. Compared to the distribution of results with 16 trials, the distribution with 50 trials j1EopTmnsJsYHFr1WikKhp5fyU2aEd59JcIqxTJvzQI= resembles the normal curve, because with more possible outcomes, the distribution is L4UsIC4XF2D9vWacd/4VPvEEFquRFD1v.

2
Try again.
Incorrect. See above for the correct answers.
Great job.
Now increase the number of trials to 50. According to the applet, the most likely result will be that ____ of the tosses will come up heads. Compared to the distribution of results with 16 trials, the distribution with 50 trials ____ resembles the normal curve, because with more possible outcomes, the distribution is ____.

### Question 1.3

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
The most likely outcome for any given probability p and number of trials n is p times n, so the most likely outcomes are that you will toss 2, 5, and 7 heads if the probability of heads is 0.2, 0.5, or 0.7 respectively.