A coin was tossed n 1000 times, and the proportion of heads observed was 0.51. Do we have evidence to conclude that the coin is unfair?

My approach:

expected value for the number of heads is 1000*0.15=510

in theory, the probability will be 0.5 and that means 500 out of 1000 coins would show head

so this is fair

  • $\begingroup$ The standard way to talk about this is with a hypothesis test. Did you learn about that in statistics? $\endgroup$
    – Mark
    Feb 28, 2017 at 3:51
  • $\begingroup$ Yes, so should I do a hypothesis test that null hypothesis is p(head)=0.5 and another hypothesis that P(head) is not 0.5? $\endgroup$
    – max
    Feb 28, 2017 at 3:53
  • $\begingroup$ Yes. Choose a p-value beforehand such that you would be willing to reject the null hypothesis. Answer this question: What is the probability that a fair coin would be more than 0.01 away from fair when tossed 1000 times? You might have also learned that you can use a normal distribution as an approximation when the number of tosses is this high. $\endgroup$
    – Mark
    Feb 28, 2017 at 3:56
  • $\begingroup$ I think you meant to to look at '1000(.51) = 510'. $\endgroup$
    – BruceET
    Feb 28, 2017 at 20:26

1 Answer 1


With a sample as large as $n = 1000$ and $p$ in the vicinity of 1/2, a normal approximation should work well.

To test the null hypothesis $H_0: p = .5$ against the alternative $H_a: p > .5$ you need to compute the test statistic $$Z = \frac{\hat p - p_0}{\sqrt{p_0(1-p_0)/n}}.$$

In your example, you have $\hat p = 0.51,\,p_0 = 0.5,$ and $n = 1000.$ You would reject the null hypothesis $H_0,$ saying that the data from the $n = 1000$ flips of the coin are inconsistent with the behavior of a fair coin, at the 5% level if $Z > 1.645.$ (The 'critical value' cutting 5% of the probability from the upper tail of a standard normal distribution, can be found from printed normal CDF tables or from software.)

I will leave it to you to finish this. Please compare this with what is in your textbook, and leave a Comment if you still need help.

Notes: It would not be unusual to get 510 or more Heads in 1000 tosses of a fair coin. That would happen over 25% of the time. A truly unusual outcome would be to get exactly 500 Heads in 1000 tosses (probability about 0.025). Results from R statistical software:

1 - pbinom(509, 1000, .5)
## 0.2739864                # P(510 Heads or more)
dbinom(500, 1000, .5)
## 0.02522502               $ P(EXACTLY 500 heads)
  • $\begingroup$ I got the z value as 0.6329 and the p-value =0.528 and therefore we cannot reject the null hypothesis. Does it mean that the coin is not fair? I have the same p-value in the back of book $\endgroup$
    – max
    Mar 1, 2017 at 3:15
  • 1
    $\begingroup$ You can look at it two ways: (a) It means the data we have is consistent with a fair coin. (b) It means you don't have solid evidence the coin is biased. We will never know 'for sure' whether the coin is biased. We can only see our data, and make statements about our data. And we don't have much of it. To try to make reasonable judgments about the difference between $p = .50$ and $p=.51,$ it would take about $n=10,000$ tosses of the coin. $\endgroup$
    – BruceET
    Mar 1, 2017 at 4:43
  • $\begingroup$ BTW: From what you say about the answer in the book, it seems they are testing $H_0: p = .5$ against the two-tailed alternative $H_a: p \ne .5.$ That is a reasonable choice. In my answer, notice that I assumed a right tailed alternative. $\endgroup$
    – BruceET
    Mar 1, 2017 at 5:25
  • $\begingroup$ yeah, i forgot to mention that the other hypothesis is p!=0.5 $\endgroup$
    – max
    Mar 1, 2017 at 5:36
  • $\begingroup$ Main difference then: Reject for $|Z| > 1.96$, not $Z > 1.645.$ So still can't reject. $\endgroup$
    – BruceET
    Mar 1, 2017 at 17:36

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