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Chapter 5: Problem 71

Imagine flipping a fair coin many times. Explain what should happen to the proportion of heads as the number of coin flips increases.

Short Answer

Expert verified
The proportion of heads should get closer to 0.5 (or 50%) as the number of coin flips increases because of the law of large numbers, but it does not guarantee an exact 50% distribution of heads and tails in every series of coin flips due to the randomness of each individual flip.

Step by step solution

01

Understanding the concept of probability

In case of a fair coin, there are only two outcomes: heads or tails. Thus, the probability of getting a head on each flip is \( \frac{1}{2} \) or 0.5 because each flip is an independent event.
02

Understanding the law of large numbers

The law of large numbers states that as we increase the number of experiments (in this case, coin flips), our experimental probability (observed ratio of heads) gets closer and closer to the theoretical probability (expected ratio of heads, which is 0.5 in this case).
03

Applying the concept to the problem

So, as the number of coin flips increases, the proportion of heads to total flips is expected to get closer to the theoretical probability, which is 0.5. However, it's important to remember that this doesn't mean every coin flip series will have exactly a half ratio of heads and tails. There will still be variance, but it's expected to get less as the number of coin flips increases.

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Most popular questions from this chapter

If you flip a fair coin repeatedly and the first four results are tails, are you more likely to get heads on the next flip, more likely to get tails again, or equally likely to get heads or tails?

Construct a two-way table with 80 men and 100 women in which both groups show an equal percentage of right-handedness.

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A soda-bottling plant has a flaw in that \(20 \%\) of the bottles it fills do not have enough soda in them. The sodas are sold in six-packs. Follow these steps to carry out a simulation to find the probability that three or more bottles in a six-pack will not have enough soda. a. Identify the action with a random outcome, and explain how you will simulate this outcome using the random number table in Appendix A. If you want to get the same answers we got, use all the possible one digit numbers \((0,1,2,3,4,5,6,7,8\), and 9\()\), and use some at the beginning of the list of numbers to represent bad and the rest to represent good. What numbers would represent bad and what numbers would represent good, and why? b. Describe how you will simulate a single trial. c. Describe the event of interest - that is, the event for which you wish to estimate a probability. d. Carry out 10 trials, beginning with the first digit on line 15 of the random number table in Appendix A. For each trial, list the digits chosen, the outcomes they represent, and whether or not the event of interest occurred. e. What is the experimental probability that you get three or more "bad" bottles in a six-pack?

a. How many outcomes are in the sample space? b. Assuming all of the outcomes in the sample space are equally likely, find each of the probabilities: i. all tails in 4 tosses ii. only 1 tail in 4 tosses iii. at most 1 tail in 4 tossesThe sample space given here shows all possible sequences for tossing a fair coin 4 times. The sequences have been organized by the number of tails in the sequence.
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